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UNIVERSITY  OF  CALIFORNIA. 

GIFT  OF 

WILLIAM  GILMAN  THOMPSON. 

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OUTLINES 


OF 


MATHEMATICAL  SCIENCE 


FOE  THE  SCHOOL-ROOM. 


BY 

CHARLES  DAVIES,  LL.D., 

AUTHOR    OF    A    FULL    COURSE    OF    MATHEMATICS. 


NEW  YORK: 

PUBLISHED   BY   A.    S.    BARNES   &   CO., 

Ill  &  113  William  Street  (cor.  John). 

1867. 


ADVEETISEMENT. 


TnE  attention  of  Teachers  is  respectfully  invited  to  the  Revised 
Editions  of 

Jafrhs'  ^riijntuiitd   Serin 

FOE  SCHOOLS  AND  ACADEMIES. 


1.  DAVIES'  PRIMARY  ARITHMETIC. 

2.  DAVIES'  INTELLECTUAL  ARITHMETIC. 

3.  DAVIES'  PRACTICAL  ARITHMETIC. 

4.  DAVIES'  UNIVERSITY  ARITHMETIC. 

5.  DAVIES'  PRACTICAL  MATHEMATICS. 


The  above  Works,  by  Charles  Davies,  LL.D.,  Author  of  a  Com- 
plete Course  of  Mathematics,  are  designed  as  a  full  Course  of  Arith- 
metical Instruction  necessary  for  the  practical  duties  of  business  life ; 
and  also  to  prepare  the  Student  for  the  more  advanced  Series  of  Mathe- 
matics by  the  same  Author. 

The  following  New  Editions  of  Algebra,  by  Professor  Davtes,  are 
commended  to  the  attention  of  Teachers : 

1.  DAVIES'  NEW  ELEMENTARY  ALGEBRA  AND  KEY. 

2.  DAVIES'  UNIVERSITY  ALGEBRA  AND  KEY. 

3.  DAVIES'  BOURDON'S  ALGEBRA  AND  KEY. 


Entered  according  to  Act  of  Congress,  in  the  year  one  thousand  eight  hundred 
and  sixty-seven, 

By    CHARLES    DAVIES, 

Iu  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 


PREFACE   AND   PLAN. 


-  The  object  of  this  work  is  to  furnish  to  the  teacher, 
in  the  school-room,  an  aid  and  guide  in  his  daily 
labors  of  giving  Mathematical  Instruction. 

The  work  is  but  an  analysis,  in  an  abridged  form, 
of  that  system  of  mathematical  instruction  which  has 
been  steadily  pursued  at  the  Military  Academy  over 
a  third  of  a  century,  and  which  has  given  to  that 
institution  its  celebrity  as  a  school  of  mathematical 
science. 

It  is  the  essence  of  this  system  that  a  principle  be 
taught  before  it  is  applied  to  practice — that  Science 
should  precede  Art — and  that  general  laws  and  gen- 
eral principles  are  the  only  true  foundations  of 
knowledge.  Where,  then,  for  the  instruction  of 
the  young,  can  we  find  these  laws  and  principles? 

The  first  impressions  which  the  child  receives  of 
Number  and  Space  are  the  foundations  of  his  mathe- 
matical knowledge.  They  form,  as  it  were,  a  part  of 
his  intellectual  being.  The  laws  of  mathematical 
science  are  generalized  truths  derived  from  the  con- 

1* 


PREFACE     AND    PLAN. 


sideration  of  Number  and  Space.  All  the  processes 
of  inquiry  and  investigation  are  conducted  according 
to  fixed  laws,  and  form  a  science ;  and  every  new 
thought  and  higher,  impression  form  additional  links 
in  the  lengthening  chain.  Beginning  with  first 
principles  and  elementary  combinations,  and  guided 
by  simple  laws,  lie  will  go  forward  from  the  exercises 
of  Mental  Arithmetic  to  the  higher  analysis  of 
Mathematical  Science  on  an  ascent  so  gentle,  and 
with  a  progress  so  steady,  as  scarcely  to  note  the 
changes.  And,  indeed,  why  should  he  ?  For  all 
mathematical  processes  are  alike  in  their  nature — are 
governed  by  the  same  laws,  exercise  the  same  faculties, 
and  lift  the  mind  towards  the  same  eminence. 

The  educator  regards  mathematical  science  as  the 
great  means  of  accomplishing  his  work.  The  defi- 
nitions present  clear  and  separate  ideas,  which  the 
mind  readily  apprehends.  The  axioms  afford  the 
simplest  exercises  of  the  reasoning  faculty,  and  the 
most  satisfactory  results  in  the  early  use  and  employ- 
ment of  that  faculty.  The  trains  of  reasoning  which 
follow,  are  combinations,  according  to  logical  rules, 
of  what  has  been  previously  fully  comprehended ;  and 
the  mind  and  the  argument  grow  together,  so  that 
the  thread  of  science  and  the  warp  of  the  intellect 
entwine  themselves,  and  become  inseparable.  Such 
a    training    will    lay    the    foundation    of    systematic 


PREFACE     AND     PLAN 


knowledge,     so    greatly    preferable     to     conjectural 
judgments. 

"With  these  general  views,  and  under  a  firm  con- 
viction that  mathematical  science  must  become  the 
great  basis  of  education,  I  have  bestowed  much  time 
and  labor  on  its  analysis,  as  a  subject  of  knowledge. 
I  have  endeavored  to  present  its  elements  separately, 
and  in  their  connections ;  to  point  out  and  note  the 
mental  faculties  which  it  calls  into  exercise ;  to  show 
why  and  how  it  develops  those  faculties ;  and  in 
what  respect  it  gives  to  the  whole  mental  machinery 
greater  power  and  certainty  of  action  than  can  bo 
attained  by  other  studies. 

Mathematical  knowledge  differs  from  every  other 
kind  of  knowledge  in  this  :  it  is,  as  it  were,  a  web  of 
connected  principles  spun  out  from  a  few  abstract 
ideas,  until  it  has  become  one  of  the  great  means  of 
intellectual  development  and  of  practical  utility. 
And  if  I  am  permitted  to  extend  the  figure,  I  may 
add,  that  the  web  of  the  spider,  though  perfectly 
simple,  if  we  see  the  end  and  understand  the  way  in 
which  it  is  put  together,  is  yet  too  complicated  to  be 
unravelled,  unless  we  begin  at  the  right  point,  and  ob- 
serve the  law  of  its  formation.  So  with  mathematical 
science.  It.  is  evolved  from  a  few — a  very  few — 
elementary   and    intuitive  principles;  the  law  of  its 


6  PEEFACE     AND     PLAN. 

evolution  is  simple  but  exacting,  and  to  begin  at 
the  right  place  and  proceed  in  the  right  way,  is  all 
that  is  necessary  to  make  the  subject  easy,  inter- 
esting, and   useful. 

I  have  endeavored  to  point  out  the  place  of  be- 
ginning, and  to  indicate  the  way  to  the  maUiemat- 
ical  student.  I  am  aware  that  he  is  starting  on  a 
road  where  the  guide-boards  resemble  each  other, 
and  where,  for  the  want  of  careful  observation,  they  are 
often  mistaken ;  I  have  sought,  therefore,  to  furnish 
him  with  the  maps  and  guide-books  of  an  old 
traveller. 

The  leading  idea,  in  the  construction  of  the  work, 
has  been,  to  afford  substantial  aid  to  the  professional 
teacher.  The  nature  of  his  duties  —  their  inherent 
difficulties — the  perplexities  which  meet  him  at  every 
step — the  want  of  sympathy  and  support  in  his  hours 
of  discouragement — (and  they  are  many) — are  circum- 
stances which  awaken  a  lively  interest  in  the  hearts 
of  all  who  have  shared  the  toils,  and  been  them- 
selves laborers  in  the  same  vineyard.  He  takes  his 
place  in  the  schoolhouse,  by  the  roadside,  and  there, 
removed  from  the  highways  of  life,  spends  his  days 
in  raising  the  feeble  mind  of  childhood  to  strength — 
in  planting  aright  the  seeds  of  knowledge — in  curb- 
ing the    turbulence  of   passion — in    eradicating    evil 


PREFACE     AND     PLAN 


and  inspiring  good.  The  fruits  of  his  labors  are 
seen  but  once  in  a  generation.  The  boy  must  grow 
to  manhood  and  the  girl  become  a  matron  before  he 
is  certain  that  his  labors  have  not  been  in  vain. 

Yet.  to  the  teacher  is  committed  the  high  trust  of 
forming  the  intellectual,  and,  to  a  certain  extent,  the 
moral  development  of  a  people.  He  holds  in  his 
hands  the  keys  of  knowledge.  If  the  first  moral 
impressions  do  not  spring  into  life  at  his  bidding, 
he  is  at  the  source  of  the  stream,  and  gives  direc- 
tion to  the  current.  Although  himself  imprisoned 
in  the  schoolhouse,  his  influence  and  his  teachings 
affect  all  conditions  of  society,  and  reach  over  the 
whole  horizon:  of  civilization.  He  impresses  himself 
on  the  young  of  the  age  in  which  he  lives,  and  lives 
again  in  the  age  which  succeeds  him. 

All  good  teaching  must  flow  from  copious  knowl- 
edge. The  shallow  fountain  cannot  emit  a  vigor- 
ous stream.  In  the  hope  of  doing  something  that 
may  be  useful  to  the  professional  teacher,  I  have 
attempted  a  careful  and  full  analysis  of  mathemati- 
cal science.  I  have  spread  out,  in  detail,  those  meth- 
ods which  have  been  carefully  examined  and  sub- 
jected to  the  test  of  long  experience.  If  they  are 
the  right  methods,  they  will  serve  as  standards  of 
teaching;  for,  the  principles  of  imparting  instruction 
are  the  same  for  all  branches  of  education. 


8  PREFACE     AND     PLAN. 

The  system  which  I  have  indicated  is  complete  in 
itself.  It  lays  open  to  the  teacher  the  entire  skeleton 
of  the  science — exhibits  all  its  parts  separately  and  in 
their  connection.  It  explains  a  course  of  reasoning 
simple  in  itself,  and  applicable  not  only  to  every  pro- 
cess in  mathematical  science,  but  to  all  processes  of 
argumentation  in  every  subject' of  knowledge. 

The  teacher  who  thus  combines  science  with  arl, 
no  longer  regards  Arithmetic  as  a  mere  treadmill  of 
mechanical  labor,  but  as  a  means — and  the  simplest 
means — of  teaching  the  art  and  science  of  reasoning 
on  quantity ;  and  this  is  the  logic  of  mathematics. 
If  he  would  accomplish  well  his  work,  he  must  so 
instruct  his  pupils  that  they  shall  apprehend  clearly, 
think  quickly  and  correctly,  reason  justly,  and  open 
their  minds  freely  to  the  reception  of  all  truth. 

It  may  be  proper  to  remark,  that  this  work  is  an 
abridgment,  with  some  changes  and  modifications,  of 
a  larger  work  entitled,  "  Logic  and  Utility  of  Mathe- 


FlSHKILL-ON-HUDSON, 

January,  1867. 


CONTENTS. 


SECTION    I. 

LOGIC     OF     MATHEMATICS. 

SECTION 

Definitions — Different  Kinds 1-8 

Rules  for  Constructing  Definitions 5 

Operations  of  the  Mind  concerned  in  Reasoning 6-10 

Apprehension 7 

Judgment 8 

Reasoning,  or  Discourse 9 

Language 10, 11 

Abstraction 12 

Different  Senses  of  Abstraction 13 

Generalization 14 

Singular  Terms — Common  Terms 15 

Classification 16-19 

Nature  of  Common  Terms 20 

Science 21 

Art 22 

Knowledge 23 

Facts  and  Truths 24 

Intuitive  Truths 25 

Logic  or  Reasoning 26 

Induction 27-29 

Deduction , 30 

1* 


10  CONTENTS, 


SECTIOX 

Propositions — Different  Kinds 31-35 

Syllogism 36-37 


SECTION    II. 

OUTLINES     OP     MATHEMATICAL     SCIENCE. 

Mathematics  Defined 38 

Quantity  Denned «39 

Number— Different  Kinds 40-43 

Language  of  Number 44 

Space  Denned 45, 46 

Portions  of  Space 47-50 

Parts  of  Mathematics 51 

Kinds  of  Number 52 

Language  of  Mathematics 53 

Its  Symbolical  Language 54 

Symbols  denoting  Quantities 55 

Arithmetic  Defined 56 

Geometry 57 

Algebra. .  .> 58 

Analysis 59 

Different  Branches  of  Mathematics 60 

Signs  of  Operation 61 

Remarks  on  Language 62-66 


SECTION    III. 

SCIENCE     OF     NUMBERS. 

First  Notions  of  Numbers 67 

Three  Forms  of  Language 68, 69 


CONTENTS.  11 


SECTION 

Ideas  of  Numbers  Generalized 70,  71 

Unity  and  a  Unit  Defined 72 

Alphabet  Defined , 73 

Arithmetical  Alphabet , 74 

Spelling  and  Reading  in  Addition 75 

General  Reading 76-79 

Spelling  and  Reading  in  Subtraction 80,  81 

Spelling  and  Reading  in  Multiplication 82 

Spelling  and  Reading  in  Division ". 83 

Units  increasing  by  Scale  of  Tens 84-88 

Scales — Uniform  and  Varying 89 

Units  increasing  by  Scale  of  Tens 90,  91 

Units  increasing  by  Varying  Scales 92,  93 

Integral  Units — Different  Kinds 94,  95 

Abstract  Units  Defined 96,  97 

Units  of  Currency ; ; 98-100 

Units  of  Weight 1 01-1 04 

Units  of  Time 105 

Units  of  Length 106 

Units  of  Surface 107-110 

Units  of  Volume 111-114 

Units  of  Angular  Measure 115 

Advantages  of  System  of  Unities 116 

Application  to  the  Four  Ground-Rules 117-119 

Logical  Form  for  Addition 120 

Logical  Form  for  Subtraction 121 

Forms  for  Multiplication  and  Division 122-123 

METRIC      SYSTEM. 

Naming — Measurement  of  Length 124, 125 

Square  Measure 126 

Measures  of  Volumes 127 

Dry  Measure 128 


12  CONTENTS. 


BECTIOK 

Liquid  Measure 129 

Weights 130 

Nature  of  Metric  System 131 

FRACTIONAL     UNITS. 

Scale  of  Tens » 132-134 

Fractional  Units  in  General 135-137 

Advantages  of  Fractional  Units 138, 139 

RATIO     AND     PROPORTION. 

Ratio  Defined,  and  Properties 140-144 


SECTION    IT. 


GEOMETRY. 


Geometry — Defined,  Demonstration 145 

Geometrical  Magnitudes 146 

Lines — Straight  and  Curved 147 

Surfaces — Plain  and  Curved 148 

Plane  Figures — Different  Kinds 149-153 

Volumes — Different  Kinds 154 

Angles 155 

General  Remarks 156-157 

Comparison  of  Figures 158-164 

Properties  of  Figures 165 

Marks  of  what  may  be  Proved 166 

Demonstration — Its  Nature. . .   167-170 

Two  Kinds  of  Demonstration 171-175 

Proportion  of  Figures 176 


CONTENTS.  13 


SECTION 

Proportion  Defined 177 

Inverse  Proportion 178 

Comparison  of  Figures 179 

Comparison  of  Volumes 180 

Ratio  of  Two  Magnitudes 181 

Recapitulation  for  Geometry 182 

Suggestions  to  those  who  Teach  Geometry 182 


SECTION  V. 


ANALYSIS. 


Analysis  Defined 183 

Its  General  Principles 183-187 

Principal  branches  of  Analysis 188 

Algebra 189 

Analytical  Geometry 190 

Analytical  Trigonometry 191 

Differential  and  Integral  Calculus 192-194 


SECTION   VI. 


ALGEBRA. 


General  Properties 195-196 

How  Quantities  may  be  Increased  or  Diminished 197 

Signs  Denoting  the  Operations  of  Change 198 

Symbols — their  Nature 199 

Coefficient— Its  Use 200-201 

Exponents  Its  Use 202-203 


14  CONTENTS. 


SECTION 

Division — Its  Signs 204 

Extraction  of  Roots — Signs 205 

Minus  Sign 206-207 

Subtraction 208 

Multiplication 209 

Rules  for  the  Signs 209-211 

Zero  and  Infinity 212-216 

Nature  of  the  Equation 217-221 

Axioms  and  Operations 221 

General  Remarks ; . . .  223 

Suggestions  to  those  who  Teach  Algebra 225 


SECTION  I. 

LOGIC   OF   MATHEMATICS. 


DEFINITIONS  —  OPERATIONS     OP     THE     MIND  —  TERMS  —  LOGIC  — 
INDUCTIVE  —  DEDUCTrVE  —  SYLLOGISM. 

•       DEFINITIONS. 

§  1.  Definition  is  a  metaphorical  word,  which  lit- 
erally signifies  "laying  down  a  boundary."  All  de- 
finitions are  of  names,  and  of  names  only;  but  in 
some  definitions,  it  is  clearly  apparent,  that  nothing 
is  intended  except  to  explain  the  meaning  of  the 
word;  while  in  others,  besides  explaining  the  mean- 
ing of  the  word,  it  is  also  implied  that  there  exists, 
or  may  exist,  a  thing  corresponding  to  the  word. 

§  2.  Definitions  which  do  not  imply  the  existence 
of  things  corresponding  to  the  words  defined,  are 
those  usually  found  in  the  Dictionary  of  one's  own 
language.  They  explain  only  the  meaning  of  the 
word  or  term,  by  giving  some  equivalent  expression 
which  may  happen  to  be  better  known.    Definitions 


16  MATHEMATICS. 

which  imply  the  existence  of  things  corresponding 
to  the  words  defined,  do  more  than  this. 

For  example:  "A  triangle  is  a  rectilineal  figure 
having  three  sides."     This  definition  does  two  things: 

1st.  It  explains  the  meaning  of  the  word  triangle; 
and, 

2d.  It  implies  that  there  exists,  or  may  exist,  a  rec- 
tilineal figure  having  three  sides. 

§  3.  To  define  a  word  when  the  definition  is  to  im- 
ply the  existence  of  a  thing,  is  to  select  from  all  the 
properties  of  the  thing  those  which  are  most  simple, 
general,  and  obvious ;  and  the  properties  must  be  very 
well  known  to  us  before  we  can  decide  which  are  the 
fittest  for  this  purpose.  Hence,  a  thing  may  have 
many  properties  besides  those  which  are  named  in 
the  definition  of  the  word  which  stands  for  it. 

§  4.  In  Mathematics,  and  indeed  in  all  exact  sci- 
ences, names  imply  the  existence  of  the  things  which 
they  name ;  and  the  definitions  of  those  names  express 
attributes  of  the  things ;  so  that  no  correct  definition 
whatever,  of  any  mathematical  term,  can  be  devised, 
which  shall  not  express  certain  attributes  of  the  thing 
corresponding  to  the  name.  Every  definition  of  this 
class  is  a  tacit  assumption  of  some  proposition  which 
is  expressed  by  means  of  the  definition,  and  which 
gives  to  such  definition  its  importance. 


ITS    LOGIC.  17 


§  5.  All  the  reasonings  in  mathematics,  which  rest 
ultimately  on  definitions,  do,  in  fact,  rest  on  the  intu- 
itive inference,  that  things  corresponding  to  the  words 
defined  have  a  conceivable  existence  as  subjects  of 
thought,  and  do  or  may  have  proximately,  an  actual 
existence.* 

OPERATIONS    OF   THE    MIND    CONCERNED    IN    REASONING. 

§  6.  There  are  three  operations  of  the  mind  which 
are  immediately  concerned  in  reasoning. 

1st.  Simple  Apprehension  ;  2d.  Judgment ;  3d.  Rea- 
soning or  Discourse. 

§  7.  Simple  apprehension  is  the  notion  (or  concep- 
tion) of  an  object  in  the  mind,  analogous  to  the  per- 
ception of  the  senses.  It  is  either  Incomplex  or 
Complex.  Incomplex  Apprehension  is  of  one  object, 
or  of  several  without  any  relation  being  perceived 
between  them,  as  of  a  triangle,  a  square,  or  a  circlet 
Complex  is  of  several  with  such  a  relation,  as  of  a 
triangle  within  a  circle,  or  a  circle  within  a  square. 

§  8.  Judgment  is  the  comparing  together  in  the 
mind  two  of  the  notions  (or  ideas)  which  are  the  ob- 

*  There  are  four  rules  which  aid  us  in  framing  definitions. 
1st.  The  definition  must  be  adequate  :    that  is,  neither  too  ex- 
tended, nor  too  narrow  for  the  word  defined. 


18  MATHEMATICS, 


jects  of  apprehension,  whether  complex  or  ineomplex, 
and  pronouncing  that  they  agree  or  disagree  with  each 
other,  or  that  one  of  them  belongs  or  does  not  helong 
to,  the  other:  for  example:  that  a  right-angled  tri- 
angle and  an  equilateral  triangle  belong  to  the  class 
of  figures  called  triangles ;  or.  that  a  square  is  not 
a  circle.  Judgment,  therefore,  is  either  Affirmative 
or  Negative. 

§  9.  Reasoning  (or  discourse)  is  the  act  of  proceeding 
from  certain  judgments  to  another  foun ded  upon  them 
(or  the  result  of  them). 

§  10.  Language  affords  the  signs  by  which  these 
operations  of  the  mind  are  recorded,  expressed,  and 
communicated.  It  is  also  an  instrument  of  thought, 
and  one  of  the  principal  helps  in  all  mental  operations ; 
hence,  any  imperfection  in  the  instrument,  or  in  the 
mode  of  using  it,  will  materially  affect  any  result  at- 
tained through  its  aid. 


2d.  The  definition  must  be  in  itself  plainer  than  the  word  de- 
fined, else  it  would  not  explain  it. 

3.  The  definition  should  be  expressed  in  a  convenient  number  of 
appropriate  words. 

4th.  When  the  definition  implies  the  existence  of  a  thing  cor- 
responding to  the  word  defined,  the  certainty  of  that  existence 
must  be  intuitive. 


ITS    LOGIC.  19 


§  11.  Every  branch  of  knowledge  has,  to  a  certain 
extent,  its  own  appropriate  language  ;  and  for  a  mind 
not  previously  versed  in  the  meaning  and  right  use  of 
the  various  words  and  signs  which  constitute  the  lan- 
guage, to  attempt  the  study  of  methods  of  philosophiz- 
ing, would  be  as  absurd  as  to  attempt  reading  before 
learning  the  alphabet. 


AB  8  TK  ACTION. 

§  12.  The  faculty  of  abstraction  is  that  power  of  the 
mind  which  enables  us,  in  contemplating  any  object  (or 
objects),  to  attend  exclusively  to  some  particular  cir- 
cumstance belonging  to  it,  and  quite  withhold  our  at- 
tention from  the  rest.  Thus,  if  a  person,  in  contem- 
plating a  rose  should  make  the  scent  a  distinct  object 
of  attention,  and  lay  aside  all  thought  of  the  form, 
color,  &c,  lie  would  draw  off,  or  abstract  that  par- 
ticular part ;  and  therefore  employ  the  faculty  of 
abstraction.  He  would  also  employ  the  same  faculty 
in  considering  whiteness,  softness,  virtue,  existence,  as 
entirely  separate  from  particular  objects. 

§  13.  The  term  abstraction,  is  also  used  to  denote  the 
operation  of  abstracting  from  one  or  more  things  the 
particular  part  under  consideration;  and  likewise  to 
designate  the  state  of  the  mind  when  occupied  by  ab- 


20  MATHEMATICS. 

stract    ideas.     Hence,    abstraction    is    used    in   three 
senses  : 

1st.    To  denote  a  facility  or  power  of  the  mind; 

2d.   To  denote  a  process  of  the  mind ;   and, 

3d.    To  denote  a  state  of  the  mind. 


GENERALIZATION. 

§  14.  Generalization  is  the  process  of  contemplating 
the  agreement  of  several  objects  in  certain  points 
(that  is,  abstracting  the  circumstances  of  agreement, 
disregarding  the  differences),  and  giving  to  all  and 
each  of  these  objects  a  name  applicable  to  them  in 
respect  to  this  agreement.  For  example;  we  give  the 
name  of  triangle,  to  every  rectilineal  figure  having 
three  sides  ;  thus  we  abstract  this  property  from  all  the 
others  (for,  the  triangle  has  three  angles,  may  be  equi- 
lateral, or  scalene,  or  right-angled),  and  name  the  en- 
tire class  from  the  property  so  abstracted.  Generaliza- 
tion therefore  necessarily  implies  abstraction;  though 
abstraction  does  not  imply  generalization. 


TERMS  —SINGULAR   TERMS — COMMON   TERMS. 

§  15.  An  act  of  apprehension,  expressed  in  language, 
is  called  a  Term.  Proper  names,  or  any  other  terms 
which  denote  each  but  a  single  individual,  as  "  Caesar," 
"  the  Hudson,"  &c,  are  called  Singular  Terms. 


ITS    LOGIC.  21 


On  the  other  hand,  those  terms  which  denote  any 
individual  of  a  whole  class  (which  are  formed  by  the 
process  of  abstraction  and  generalization),  are  called 
Common  or  general  Terms.  For  example ;  quadrilat- 
eral is  a  common  term,  applicable  to  every  rectilineal 
plane  figure  having  four  sides;  Eiver,  to  all  rivers; 
and  Conqueror,  to  all  conquerors.  The  individuals,  for 
which  a  common  term  stands,  are  called  its  Slgni- 
ficates. 


CLASSIFICATION.  * 

§  16.  Common  terms  afford  the  means  of  classifica- 
tion ;  that  is,  of  the  arrangement  of  objects  into 
classes,  with  reference  to  some  common  and  distin- 
guishing characteristic.  A  collection,  comprehending 
a  number  of  objects,  so  arranged,  is  called  a  Genus  or 
Species — genus  being  the  more  extensive  term,  and 
often  embracing  many  species. 

For  example  :  animal  is  a  genus  embracing  every 
thing  which  is  endowed  with  life,  the  power  of  volun- 
tary motion,  and  sensation.  It  has  many  species,  such 
as  man,  beast,  bird,  &c.  .If  we  say  of  an  animal,  that 
it  is  rational,  it  belongs  to  the  species  man,  for  this  is 
the  characteristic  of  that  species.  If  we  say  that  it  has 
wings,  it  belongs  to  the  species  bird,  for  this,  in  like 
manner,  is  the  characteristic  of  the  species  bird. 

A  species  may  likewise  be  divided  into  classes  or 


22  MATHEMATICS. 

subspecies ;  thus  the  species  man,  may  be  divided  into 
the  classes,  male  and  female,  and  these  classes  may  be 
again  divided  until  we  reach  the  individuals. 

§  IT.  Now,  it  will  appear  from  the  principles  which 
govern  this  system  of  classification,  that  the  character- 
istic of  a  genus  is  of  a  more  extensive  signification, 
but  involves  fewer  particulars  than  that  of  a  species. 
In  like  manner,  the  characteristics  of  a  species  is  more 
extensive,  but  less  full  and  complete,  than  that  of  a 
subspecies  or  class,  and  the  characteristics  of  these  less 
full  than  that  of  an  individual.  ^^^ 

For  example ;  if  we  take  as  a  genus  the    /   1    \ 
Quadrilaterals   of    Geometry,   of  which   the 

characteristic  is,   that  they  have  four  sides,     / v 

then  every  plane  rectilineal  figure,  having  /     2     \ 
four    sides,    will   fall    under    this    class.     If, 
then,  we  divide  all  quadrilaterals   into  two  V     ~    \ 
species,  viz.  those  whose  opposite  sides,  taken     * — - — A 
two   and   two,   are   not   parallel,  and    those 
whose  opposite  sides,  taken  two  and  two,  are 
parallel,  we  shall  have  in  the  first  class,  all 
irregular  quadrilaterals,  including  the  trape- 
zoid (1  and  2) ;  and  in  the  other,  the  parallel- 
ogram, the  rhombus,  the  rectangle,  and  the 
square  (3,  4,  5,  and  6). 

If,  then,  we  divide  the  first  species  into 
two  subspecies  or  classes,  we  shall  have  in 


ITS    LOGIC.  23 


the  one,  the  irregular  quadrilaterals,  called  trapeziums 
(1),  and  in  the  other,  the  trapezoids  (2);  and  each  of 
these  classes,  being  made  up  of  individuals  having  the 
same  characteristics,  are  not  susceptible  of  further 
division. 

If  we  divide  the  second  species  into  two  classes,  ar- 
ranging those  which  have  oblique  angles  in  the  one, 
and  those  which  have  right  angles  in  the  other,  we 
shall  have  in  the  first,  two  varieties,  viz.  the  common 
parallelogram  (3),  and  the  equilateral  parallelogram  or 
rhombus  (4) ;  and  in  the  second,  two  varieties  also,  viz. 
the  rectangle  and  the  square  (5  and  6). 

Now,  each  of  these  six  figures  is  a  quadrilateral; 
and  hence,  possesses  the  characteristic  of  the  genus ; 
and  each  variety  of  both  species  enjoys  all  the  charac- 
teristics of  the  species  to  which  it  belongs,  together 
with  some  other  distinguishing  feature  of  a  species 
nearer  the  genus  ;  and  similarly,  of  all  classifications. 

§  18.  In  special  classifications,  it  is  often  not  neces- 
sary to  begin  with  the  most  general  characteristics ; 
and  then  the  genus  with  which  we  begin,  is  in  fact  but 
a  species  of  a  more  extended  classification,  and  is  called 
a  Subaltern  Genus. 

For  example;  if  wre  begin  with  the  genus  Paral- 
lelogram, we  shall  at  once  have  two  species,  viz.  those 
parallelograms  whose  angles  are  oblique  and  those 
whose  angles  are  right  angles ;   and  in   each  species 


24  MATHEMATICS. 


there  will  be  two  varieties,  viz.  in  the  first,  the  common 
parallelogram  and  tlie  rhombus  ;  and  in  the  second,  the 
rectangle  and  square. 

§  19.  A  genus  which  cannot  be  considered  as  a  spe- 
cies, that  is,  which  cannot  be  referred  to  a  more  ex- 
tended classification,  is  called  the  highest  genus ;  and  a 
species  which  cannot  be  considered  as  a  genus,  because 
it  contains  only  individuals  having  the  same  character- 
istics, is  called  the  lowest  species. 


NATURE     OF     COMMON     TEEMS. 

§  20.  It  should  be  steadily  kept  in  mind,  that  the 
"common  terms"  employed  in  classification,  have  not, 
as  the  names  of  individuals  have,  any  real  existing 
thing  in  nature  corresponding  to  them ;  but  that  each 
is  merely  a  name  denoting  a  certain  inadequate  notion 
which  our  minds  have  formed  of  an  individual.  But 
as  this  name  does  not  include  any  thing  wherein  that 
individual  differs  from  others  of  the  same  class,  it  is  ap- 
plicable equally  well  to  all  of  them,  as  to  any  of  them. 
Thus,  quadrilateral  denotes  no  real  thing,  distinct  from 
each  individual,  but  merely  any  rectilineal  figure  of 
four  sides,  viewed  inadequately  /  that,  is  abstracting 
and  omitting  all  that  is  peculiar  to  each  individual  of 
the  class.     By  this  means,  a  common  term  becomes 


ITS    LOGIC.  25 


applicable  alike  to  any  one  of  several  individuals ;  or, 
taken  in  the  plural,  to  several  individuals  together. 

In  regard  to  classification,  we  should  also  bear  in 
mind,  that  we  may  fix,  arbitrarily,  on  the  characteristic 
which  we  choose  to  abstract  and  consider  as  the  basis 
of  our  classification,  disregarding  ail  the  rest :  so  that 
the  same  individual  may  be  referred  to  any  of  several 
different  species,  and  the  same  species  to  several  genera, 
as  suits  our  purpose. 

SCIENCE. 

§  21.  Science,  in  its  popular  signification,  means 
knowledge.  In  a  more  restricted  sense,  it  means 
knowledge  reduced  to  order  ;  that  is,  knowledge  so 
classified  and  arranged  as  to  be  easily  remembered, 
readily  referred  to,  and  advantageously  applied.  In  a 
more  strict  and  technical  sense,  it  has  another  significa- 
tion. 

"Every  thing  in  nature,  as  well  in  the  inanimate  as 
in  the  animated  world,  happens  or  is  done  according  to 
rules,  though  we  do  not  always  know  them.  Water 
falls  according  to  the  laws  of  gravitation,  and  the  mo- 
tion of  walking  is  performed  by  animals  according  to 
rules.  The  fish  in  the  water,  the  bird  in  the  air,  move 
according  to  rules.  There  is  nowhere  any  want  of 
rule.  When  we  think  we  find  that  want,  we  can  only 
say  that,  in  this  case,  the  rules  are  unknown  to  us," 

2 


26  MATHEMATICS 


Assuming  tliat  all  the,  p] ]  en  omen  a  of  nature  are  con- 
sequences of  general  and  immutable  laws,  we  may  de- 
fine Science  to  be  the  analysis  of  those  laws — compre- 
hending not  only  the  connected  processes  of  experiment 
and  reasoning  which  make  them  known  to  man,  but 
also  those  processes  of  reasoning  which  make  known 
their  individual  and  concurrent  operation  in  the  devel- 
opment of  individual  phenomena. 


ART. 

§  22.  Art  is  the  application  of  knowledge  to  practice. 
Science  is  conversant  about  knowledge  :  Art  is  the  use 
or  application  of  knowledge,  and  is  conversant  about 
works.  Science  has  knowledge  for  its  object :  Art  has 
knowledge  for  its  guide.  A  principle  of  science,  when 
applied,  becomes  a  rule  of  art.  The  developments  of 
science  increase  knowledge :  the  applications  of  art  add 
to  works.  Art,  necessarily,  presupposes  knowledge: 
art,  in  any  but  its  infant  state,  presupposes  scientific 
knowledge;  and  if  every  art  does  not  bear  the  name  of 
the  science  on  which  it  rests,  it  is  only  because  several 
sciences  are  often  necessary  to  form  the  groundwork  of 
a  single  art.  Such  is  the  complication  of  human 
affairs,  that  to  enable  one  thing  to  be  do?ie,  it  is  often 
requisite  to  know  the  nature  and  properties  of  many 
things. 


ITS    LOGIC.  27 


KNOWLEDGE 


§  23.  Kn<>wlfxge  is  a  clear  and  certain  conception 
of  that  which  is  true,  and  implies  three  things : 

1st.  Firm  belief ;   2d.  Of  what  is  true ;   and,  3d.  On 


sufficient  grounds. 


FACTS     AND     TRUTHS. 

§  24.  Our  knowledge  is  of  two  kinds  :  of  facts  and 
truths.  A  fact  is  any  thing  that  has  been  or  is.  That 
the  sun  rose  yesterday,  is  a  fact:  that  he  gives  light 
to-day,  is  a  fact.  That  water  is  fluid  and  stone  solid, 
are  facts.  We  derive  our  knowledge  of  facts  through 
the  medium  of  the  senses. 

Truth  is  an  exact  accordance  with  what  has  bekn, 
is,  or  shall  be.  There  are  two  methods  of  ascertain- 
ing truth : 

1st.  By  comparing  known  facts  with  each  other; 
and, 

2dly.  By  comparing  known  truths  with  each  other. 

Hence,  truths  are  inferences  from  facts  or  other 
truths  made  by  a  mental  process  called  Reasoning. 

intuitive    truths. 

§  25.  Intuitive  Truths  are  those  which  become 
known,  by  considering  all  the  facts  on  which  they  de- 


28  MATHEMATICS. 

pend,  and  which  are  inferred  the  moment  the  facts  are 
apprehended. 

The  term  Intuition  is  strictly  applicable  only  to  that 
mode  of  contemplation  in  which  we  look  at  facts,  or 
classes  of  facts,  and  apprehend  the  relations  of  those 
facts  at  the  same  time,  and  by  the  same  act  by  which 
we  apprehend  the  facts  themselves.  Hence,  intuitive 
or  self-evident  truths  are  those  which  are  conceived  in 
the  mind  immediately;  that  is,  which  are  perfectly  con- 
ceived by  a  single  process,  the  moment  the  facts  on 
which  they  depend  are  apprehended.  They  are  neces- 
sary consequences  of  conceptions  respecting  which  they 
are  asserted.  The  axioms  of  Geometry  afford  the  sim- 
plest and  most  unmistakable  class  of  such  truths. 

"  A  whole  is  equal  to  the  sum  of  all  its  parts,"  is 
an  intuitive  or  self-evident  truth,  inferred  from  facts 
previously  learned.  For  example ;  having  learned 
from  experience  and  through  the  senses  what  a  whole 
is,  and,  from  experiment,  the  fact  that  it  may  bo 
divided  into  parts,  the  mind  perceives  the  relation 
between  the  whole  and  the  sum  of  the  parts,  viz.  that 
they  are  equal ;  and  then,  by  the  reasoning  process, 
infers  that  the  same  will  be  true  of  every  other  thing ; 
and  hence,  pronounces  the  general  truth,  that  "  a  whole 
is  equal  to  the  sum  of  all  its  parts/'  Here  all  the 
facts  from  which  the  conclusion  is  drawn,  are  pre- 
sented to  the  mind,  and  the  conclusion  is  immediately 
made ;    hence,  it  is  an  intuitive  or  self-evident  truth. 


ITS     LOGIC.  29 


All  the  other  axioms  of  Geometry  are  deduced  from 
premises  and  by  processes  of  inference,  entirely  similar. 
We  would  not  call  these  experimental  truths,  for  they 
are  not  alone  the  results  of  experiment  or  experience. 
Experience  and  experiment  furnish  the  requisite  in* 
formation,  but  the  reasoning  power  evolves  the  gen- 
eral truth. 


LOGIC     OK     REASONING. 

§  26.  Logic  takes  note  of  and  decides,  upon  the 
sufficiency  of  the  evidence  by  which  truths  are  estab- 
lished. Our  assent  to  the  conclusion  being  grounded 
on  the  truth  of  the  premises,  we  never  could  arrive  at 
any  knowledge  by  reasoning,  unless  something  were 
known  antecedently  to  all  reasoning.  It  is  the  prov- 
ince of  Logic  to  furnish  the  tests  by  which  all  truths 
that  are  not  intuitive  may  be  inferred  from  the  prem- 
ises. It  has  nothing  to  do  with  ascertaining  facts,  nor 
with  any  proposition  which  claims  to  be  believed  on  its 
own  intrinsic  evidence.  But,  so  far  as  our  knowledge 
is  founded  on  truths  made  such  by  evidence,  that  is, 
derived  from  facts  or  other  truths  previously  known, 
whether  those  truths  be  particular  truths,  or  general 
propositions,  it  is  the  province  of  Logic  to  supply  the 
tests  for  ascertaining  the  validity  of  such  evidence,  and 
whether  or  not  a  belief  founded  on  it  would  be  well 
grounded.     And  since  by  far  the  greatest  portion  of 


30  MATHEMATICS. 

our  knowledge,  whether  of  particular  or  general  truths, 
is  avowedly  matter  of  inference,  nearly  the  whole,  not 
only  of  science,  but  of  human  conduct,  is  amenable  to 
the  authority  of  Logic.  Logic  is  divided  into  two 
kinds :   Inductive  and  Deductive. 

INDUCTION. 

§  27.  That  part  of  Logic  which  infers  truths  from 
facts,  is  called  Induction.  Inductive  reasoning  is  the 
application  of  the  reasoning  process  to  a  given  number 
of  facts,  for  the  purpose  of  determining  if  what  has 
been  ascertained  respecting  one  or  more  of  the  indi- 
viduals, is  true  of  the  whole  class.  Hence,  Induction 
is  not  the  mere  sum  of  the  facts,  but  a  conclusion  drawn 
from  them. 

The  logic  of  Induction  consists  in  classing  the  facts 
and  stating  the  inference  from  them. 

§  "28.  Induction,  therefore,  is  a  process  of  inference. 
It  proceeds  from  the  known  to  the  unknown ;  and  any 
operation  involving  no  inference,  any  process  in  which 
the  conclusion  is  a  mere  fact,  and  not  a  truth,  does  not 
fall  within  the  meaning  of  the  term.  The  conclusion 
must  be  broader  than  the  premises.  The  premises  are 
facts :  the  conclusion  must  be  a  truth. 

Induction,  therefore,  is  a  process  of  generalization. 
It  is  that  operation  of  the  mind  by  which  we  infer  that 


ITS     LOGIC.  31 


what  we  know  to  be  true  in  a  particular  case  or  cases, 
will  be  true  in  all  cases  which  resemble  the  former  in 
certain  assignable  respects.  In  other  words,  Induction 
is  the  process  by  which  we  conclude  that  what  is  true 
of  certain  individuals  of  a  class,  is  true  of  the  whole 
class;  or  that  what  is  true  at  certain  times,  will  be 
true,  under  similar  circumstances,  at  all  times. 

§  29.  Induction  always  presupposes,  not  only  that 
the  necessary  observations  are  made  with  the  necessary 
accuracy,  but  also,  that  the  results  of  these  observations 
are,  so  far  as  practicable,  connected  together  by  general 
descriptions ;  enabling  the  mind  to  represent  to  itself 
as  wholes,  whatever  phenomena  are  capable  of  being  so 
represented. 

To  suppose,  however,  that  nothing  more  is  required 
from  the  conception  than  that  it  should  serve  to  con- 
nect the  observations,  would  be  to  substitute  hypothesis 
for  theory,  and  imagination  for  proof.  The  connecting 
link  must  be  some  character  which  really  exists  in  the 
facts  themselves,  and  which  would  manifest  itself  there- 
in, if  the  condition  could  be  realized  which  our  organs 
of  sense  require. 

For  example  :  Bakewell,  a  celebrated  English  cattle- 
breeder,  observed,  in  a  great  number  of  individual 
beasts,  a  tendency  to  fatten  readily,  and  in  a  great 
number  of  others  the  absence  of  this  constitution :  in 
every  individual  of  the  former  description,  he  observed 


32  MATHEMATICS. 

a  certain  peculiar  make,  though  they  differed  widely  in 
size,  color,  &c.  Those  of  the  latter  description  differed 
no  less  in  various  points,  but  agreed  in  being  of  a  dif- 
ferent make  from  the  others.  These  facts  were  his 
data;  from  which,  combining  them  with  the  general 
principle,  that  nature  is  steady  and  uniform  in  her 
proceedings,  he  logically  drew  the  conclusion  that 
beasts  of  the  specified  make  have  universally  a  peculiar 
tendency  to  fattening.  The  Induction  consisted  in  the 
generalization ;  that  is,  in  inferring,  from  all  the  data, 
that  certain  circumstances  would  be  found  in  the  whole 
class. 

DEDUCTION. 

§  30.  We  have  seen  that  all  processes  of  Keasoning, 
in  which  the  premises  are  particular  facts,  and  the 
conclusions  general  truths,  are  called  Inductions.  All 
processes  of  Reasoning,  in  which  the  premises  are  gen- 
eral truths  and  the  conclusions  particular  truths,  are 
called  Deductions.  Hence,  a  deduction  is  the  process 
of  reasoning  by  which  a  particular  truth  is  inferred 
from  other  truths  which  are  known  or  admitted.  The 
formula  for  all  deductions  is  found  in  the  Syllogism, 
<,he  parts,  nature,  and  uses  of  which  we  shall  now  pro^ 
ceed  to  explain. 


ITS    LOGIC.  33 


PROPOSITIONS. 

§  31.  A  proposition  is  a  judgment  expressed  in 
words.  Hence,  a  proposition  is  defined  logically,  "  A 
sentence  indicative :"  affirming  or  deifying ;  therefore, 
it  must  not  be  ambiguous,  for  that  which  has  more 
than  one  meaning  is  in  reality  several  propositions ; 
nor  imperfect,  nor  ungrammaiical,  for  such  expressions 
have  no  meaning  at  all. 


§  32.  Whatever  can  be  an  object  of  belief,  or  even  of 
disbelief,  must,  when  put  into  words,  assume  the  form 
of  a  proposition.  All  truth  and  all  error  lie  in  propo- 
sitions. What  we  call  a  truth,  is  simply  a  true  propo- 
sition ;  and  errors  are  false  propositions.  To  know  the 
import  of  all  propositions,  would  be  to  know  all  ques- 
tions which  can  be  raised,  and  all  matters  which  are 
susceptible  of  being  either  believed  or  disbelieved. 

§  33.  The  first  glance  at  a  proposition  shows  that  it 
is  formed  by  putting  together  two  names.  Thus,  in 
the  proposition  "  Gold  is  yellow,"  the  property  yellow 
is  affirmed  of  the  substance  gold.  In  the  proposition, 
"  Franklin  was  not  born  in  England,"  the  fact  ex- 
pressed by  the  words  lorn  in  England  is  denied  of  the 
man  Franklin. 

2* 


34  MATHEMATICS. 

§  34.  Every  proposition  consists  of  three  parts :  the 
Subject,  the  Predicate,  and  the  Copula.  The  subject 
is  the  name  denoting  the  person  or  thing  of  which 
something  is  affirmed  or  denied  :  the  predicate  is  that 
which  is  affirmed  or  denied  of  the  subject ;  and  these 
'two  are  called  the  terms  (or  extremes),  because,  logic- 
ally, the  subject  is  placed  first,  and  the  predicate  last. 
The  copula,  in  the  middle,  indicates  the  act  of  judg- 
ment, and  is  the  sign  denoting  that  there  is  an  affirma- 
tion or  denial.  Thus,  in  the  proposition,  "  The  earth 
is  round  ;"  the  subject  is  the  words  "  the  earth,"  being 
that  of  which  something  is  affirmed  :  the  predicate,  is 
the  word  round,  which  denotes  the  quality  affirmed,  or 
(as  the  phrase  is)  predicated :  the  word  is,  which  serves 
as  a  connecting  link  between  the  subject  and  the  predi- 
cate, to  show  that  one  of  them  is  affirmed  of  the  other, 
is  called  the  Copula.  The  copula  must  be  either  is,  or 
is  not,  the  substantive  verb  being  the  only  verb  recog- 
nized by  Logic.  All  other  verbs  are  resolvable,  by 
means  of  the  verb  "  to  be,"  and  a  participle  or  adjec- 
tive.    For  example : 

"  The  Romans  conquered  ;" 

the  word  "  conquered"  is  both  copula  and  predicate, 
being  equivalent  to  "  were  victorious?'' 
Hence,  we  might  write, 

"  The  Romans  were  victorious," 


ITS    LOGIC.  35 


in  which  were  is  the  copula,  and  victorious  the  predi- 
cate. 

§  35.  A  proposition  being  a  portion  of  discourse,  in 
which  something  is  affirmed  or  denied  of  something,  all 
propositions  may  be  divided  into  affirmative  and  nega- 
tive. An  affirmative  proposition  is  that  in  which  the 
predicate  is  affirmed  of  the  subject ;  as,  "  Csesar  is 
dead."  A  negative  proposition  is  that  in  which  the 
predicate  is  denied  of  the  subject ;  as,  "  Csesar  is  not 
dead."  The  copula,  in  this  last  species  of  proposition, 
consists  of  the  words  "is  not,"  which  is  the  sign  of 
negation  ;  "  is"  being  the  sign  of  affirmation. 

SYLLOGISM. 

§  36.  A  syllogism  is  a  form  of  stating  the  connection 
which  may  exist,  for  the  purpose  of  reasoning,  between 
three  propositions.  Hence,  to  a  legitimate  syllogism, 
it  is  essential  that  there  should  be  three,  and  only 
three,  propositions.  Of  these,  two  are  admitted  to  be 
true.  The  first  is  called  the  major  premise ;  the  sec- 
ond, the  minor  premise ;  and  the  third,  which  is 
proved  from  these  two,  and  is  called  the  conclusion. 
For  example : 

"  All  tyrants  are  detestable : 
Caesar  was  a  tyrant ; 
Therefore,  Ceesar  was  detestable." 


36  MATHEMATICS. 

Now,  if  the  first  two  propositions  be  admitted,  the 
third,  or  conclusion,  necessarily  follows  from  them,  and 
it  is  proved  that  Cjssar  was  detestable. 

Of  the  two  terms  of  the  conclusion,  the  Predicate 
(detestable)  is  called  the  major  term,  because  it  comes 
from  the  major  premise ;  the  Subject  (Caesar)  the 
minor  term,  because  it  comes  from  the  minor  premise  ; 
and  these  two  terms,  together  with  the  term  "  tyrant," 
are  alone  used  in  the  three  propositions  of  the  syllo- 
gism,— each  term  being  used  twice.  Hence,  every 
syllogism  has  three,  and  only  three,  different  terms. 

The  term  tyrant,  common  to  the  two  premises,  and 
with  which  both  the  terms  of  the  conclusion  were 
separately  compared,  before  they  were  compared  with 
each  other,  is  called  the  middle  term.  Therefore,  in 
the  above  syllogism, 

"  All  tyrants  are  detestable," 
is  the  major  premise,  and 

"  Csesar  was  a  tyrant," 

the  minor  premise,  "  tyrant"  the  middle  term,  and 

"  Caesar  was  detestable  " 
the  conclusion. 

§  37.  The  syllogism,  therefore,  is  a  mere  formula  for 
ascertaining  what  may,  or  what  may  not,  be  predicated 
of  a  subject.     It  accomplishes  this  end.  by  means  of 


ITS    LOGIC.  37 


two  propositions,  viz.  by  comparing  the  given  predicate 
of  the  first  (a  Major  Premise),  and  the  given  subject  of 
the  second  (a  Minor  Premise),  respectively  with  one 
and  the  same  third  term  (called  the  middle  term),  and 
thus — under  certain  conditions,  or  laws  of  the  syllogism 
— eliciting  the  truth  (conclusion),  that  the  given  predi- 
cate must  be  predicated  of  that  subject.  It  will  be 
seen  that  the  Major  Premise  always  declares,  in  a  gen- 
eral way,  such  a  relation  between  the  Major  Term  and 
the  Middle  Term  ;  and  the  Minor  Premise  declares,  in 
a  more  particular  way,  such  a  relation  between  the 
Minor  Term  and  the  Middle  Term,  as  that,  in  the  Con- 
clusion, the  Minor  Term  must  be  put  under  the  Major 
Term  ;  or  in  other  words,  that  the  Major  Term  must  be 
predicated  of  the  Minor  Term. 

In  Mathematics,  the  reasoning  is  entirely  Deductive: 
hence,  every  process,  in  proceeding  from  known  to  un- 
known truths,  may  be  reduced  to  the  form  of  the  Syl- 
logism. 


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SECTION    II. 

OUTLINE    OF   MATHEMATICAL    SCIENCE, 


§  38.  Mathematics  is  the  science  of  quantity  ;  that 
is,  the  science  which  treats  of  the  measure  of  quantity 
and  the  relations  of  quantities  to  each  other. 

Mathematics  may  be  divided  into  two  parts,  Pure 
and  Mixed. 

The  Pure  Mathematics  embraces  the  principles  of 
the  science,  and  all  explanations  of  the  processes  "by 
which  those  principles  are  derived  from  the  laws  of 
the  abstract  quantities,  Number  and  Space. 

The  Mixed  Mathematics  embraces  the  applications 
of  those  principles  to  all  investigations,  and  to  the 
solution  of  all  questions  of  a  practical  nature,  whether 
they  relate  to  abstract  or  concrete  quantity. 

§  39*  Quantity  is  any  thing  which  can  be  increased, 
diminished,  and  measured.  There  are  but  two  kinds 
of  quantity  :    Number  and  Space, 


40  MATHEMATICS. 


NUMBER. 

§  40.  A  Number  is  a  unit  or  a  collection  of  units. 
How  do  we  apprehend  the  nature  of  Numbers  ? 

First,  by  presenting  to  the  mind,  through  the  eye, 
a  single  thing,  and  calling  it  one.  Then,  presenting 
two  things,  and  naming  them  two:  then  three  things, 
and  naming  them  three  ;  and  so  on  for  other  numbers. 
Thus,  we  acquire  primarily,  in  a  concrete  form,  our 
elementary  notions  of  number,  by  perception,  com- 
parison, and  reflection :  for,  we  must  first  perceive  how 
many  things  are  numbered  ;  then  compare  what  is 
designated  by  the  word  one,  with  what  is  designated 
by  the  words  two,  three,  etc ,  and  then  reflect  on  the 
results  of  such  comparisons  until  we  clearly  apprehend 
the  difference  in  the  signification  of  the  words.  Hav- 
ing thus  acquired,  in  a  concrete  form,  our  conceptions 
of  numbers,  we  can  consider  numbers  as  separated 
from  any  particular  objects,  and  thus  form  a  conception 
of  them  in  the  abstract. 

§  41.  The  Unit  of  a  number  is  the  single  thing 
which  forms  the  base  of  the  number. 

*. 

§  42.  An  abstract  number  is  one  whose  unit  is  not 
named;  as,  one,  two,  three,  &c. 


NAT U BE    AND    LANGUAGE.  41 

§  43.  A  denominate  numbee  is  one  whose  unit  is 
named  ;  as,  one  foot,  one  pound,  one  day,  &c.  A  de- 
nominate number  which  carries  with  it  the  idea  of 
matter,  is  called  a  concrete  number. 

Hence,  there  are  three  kinds  of  numbers:  Abstract, 
Denominate,  and  Concrete. 

Five,  is  an  abstract  number — Five  yards  a  denomi- 
nate number,  and  five  pounds  a  concrete  number.  A 
concrete  number  is  always  a  denominate  number. 

LANGUAGE     OF     NUMBER. 

§  44.  The  language  of  number  is  mixed.  It  is  com- 
posed partly  of  words  taken  from  our  common  lan- 
guage, but  its  more  perfect  and  comprehensive  form 
is  found  in  the  various  combinations  of  the  ten  charac- 
ters called  figures.  These  ten  characters  are  the  alpha- 
bet of  this  language,  and  the  various  ways  in  which 
they  are  combined  wil]  be  fully  explained  under  the 
head  Arithmetic,  a  branch  of  Mathematics  which  treats 
sof  the  properties  of  numbers,  and  of  their  language 
and  their  laws. 

SPACE. 

§  45.  Space  is  indefinite  extension.  We  acquire  our 
ideas  of  it  by  observing  that  parts  of  it  are  occupied  by 
matter  or  bodies.  This  enables  us  to  attach  a  definite 
idea  to  the  word  place.      We  are  then  able  to  say, 


42  MATHEMATICS. 

intelligibly,  that  a  point  is  that  which  has  place,  or 
position  in  space,  without  occupying  any  part  of  it. 
Having  conceived  a  second  point  in  space,  we  can  un- 
derstand the  important  axiom,  "  A  straight  line  is  the 
shortest  distance  between  two  points ;"  and  this  line 
we  call  length,  or  a  dimension  of  space. 

§  46.  If  we  conceive  a  second  straight  line  to  be 
drawn,  meeting  the  first,  but  lying  in  a  direction 
directly  from  it,  we  shall  have  a  second  dimension  of 
space,  which  we  call  breadth.  If  these  lines  be  pro- 
longed, in  both  directions,  they  will  include  four  por- 
tions of  space*,  which  make  up  what  is  called  a  plane 
surface,  or  plane  :  hence,  a  plane  has  two  dimensions, 
length  and  breadth.  If  now  we  draw  a  line  on  either 
side  of  this  plane,  we  shall  have  another  dimension  of 
space,  called  thickness:  hence,  space  has  three  dimen- 
sions—length, breadth,  and  thickness. 

PORTIONS     OF     SPACE. 

§  47.  A  portion  of  space  which  has  but  one  dimen- 
sion, is  called  a  line,  and  may  be  limited  by  two  points, 
one  at  each  extremity. 

There  are  two  kinds  of  lines,  straight  and  curved. 
A  straight  line,  is  one  which  does  not  change  its  direc- 
tion between  any  two  of  its  points,  but  a  curved  line 
changes  its  direction  at  every  point. 


NATURE    AND    LANGUAGE.  43 

§  48.  A  portion  of  space  having  two  dimensions  is 
called  a  surface.  There  are  two  kinds  of  surfaces — 
Plane  Surfaces  and  Curved  Surfaces.  With  the  for- 
mer, a  straight  line,  having  two  points  in  common,  will 
always  coincide,  however  it  may  be  placed,  while  with 
the  latter  it  will  not.  The  boundaries  of  surfaces  are 
lines,  straight  or  curved. 

§  49.  A  portion  of  space  having  three  dimensions  is 
called  a  Yolume.  Volumes  are  bounded  either  by 
plane  or  curved  surf  ices. 

§  50.  A  portion  of  space  included  between  lines,  or 
between  planes,  is  called  an  angle.  Hence,  there  are 
four  kinds  of  geometrical  magnitudes :  viz.  Lines,  Sur- 
faces, Volumes,  and  Angles. 


PARTS    OF    MATHEMATICS. 

§  51.  Referring  to  the  diagram,  on  page  38,  we  have 
a  complete  map  of  Mathematical  Science.     We  see, 

1.  That  Quantity  is  divided  into  two  kinds,  Number 
and  Space. 

2.  That  number  is  divided  into  four  kinds,  Abstract 
Number,  Currency,  Weight,  and  Time. 

3.  That  space  is  a%o  divided  into  four  kinds,  Length, 
Surfaces,  Volumes,  and  Angles. 

4.  All  these  quantities  may  be  expressed  by  num- 


44  MATHEMATICS. 

bers.  Of  these  numbers,  one  only  is  abstract  and 
seven  denominate — and  of  the  seven  denominate  num- 
bers, only  one,  weight,  is  concrete. 

'  §  52.  The  class,  or  kind  of  number,  is  determined  by 
the  Unit  which  forms  its  base  :  thus, 

An  abstract  number  has  for  its  base  the  unit  one. 

A  number  in  currency  has  for  its  base  a  unit  of 
currency,  as  1  dollar,  1  dime,  1  cent ;  or  in  the  English 
currency,  1  £,  1  s.,  Id.;  or  in  the  French,  1  franc,  or 
1  Napoleon. 

In  weight,  the  unit  is  1  pound,  1  hundred,  1  ton,  or 
1  ounce  ;  or  any  other  known  unit. 

In  time,  the  unit  is  1  day,  1  year,  1  week,  1 
hour,  &c. 

In  length,  it  is  1  foot,  1  yard,  1  rod,  1  mile,  or  any 
other  known  distance. 

In  surfaces,  or  square  measure,  the  unit  is  1  square 
inch,  1  square  foot,  1  square  yard,  1  acre,  or  any  other 
known  surface. 

In  volumes,  the  unit  is  1  cubic  foot,  1  cubic  yard, 
1  gallon,  1  quart,  or  any  other  known  and  limited 
portion  of  space. 

In  angles,  the  most  common  unit  is  the  degree, 
though  in  geometry,  we  generally  compare  all  angles 
with  the  right  angle. 


NATURE  AND  LANGUAGE.        45 


LANGUAGE  OF  MATHEMATICS. 

§  53.  The  language  of  Mathematics  is  mixed.  Al- 
though composed  mainly  of  symbols,  which  are  defined 
with  reference  to  the  uses  which  are  made  of  them,  and 
therefore  have  a  precise  signification,  it  is  also  com- 
posed,  in  part,  of  words  transferred  from  our  common 
language.  The  symbols,  although  arbitrary  signs,  are, 
nevertheless,  entirely  general,  as  signs  and  instruments 
of  thought ;  and  when  the  sense  in  which  they  are  used 
is  once  fixed,  by  definition,  they  preserve  throughout 
the  entire  analysis  precisely  the  same  signification. 
The  meaning  of  the  words,  borrowed  from  our  common 
vocabulary,  is  often  modified,  and  sometimes  entirely 
changed,  when  the  words  are  transferred  to  the  lan- 
guage of  science.  They  are  then  used  in  a  particular 
sense,  and  are  said  to  have  a  technical  signification. 

It  is  of  the  first  importance  that  the  elements  of  the 
language  be  clearly  understood, — that  the  signification 
of  every  word  or  symbol  be  distinctly  apprehended, 
and  that  the  connection  between  the  thought  and  the 
word  or  symbol  "which  expresses*  it,  be  so  well  estab- 
lished that  the  one  shall  immediately  suggest  the  other. 

THE     SYMBOLICAL    LANGUAGE. 

§  54.  The  Symbolical  language  of  Mathematics  is 
divided  into  two  parts  : 


46  MATHEMATICS, 


1.  The  symbols  which  denote  quantities  ;  and 

2.  The  symbols  which  denote  operations  to  be  per- 
formed on  quantities. 


SYMBOLS    DENOTING    QUANTITIES. 

§  55.  There  are  three  classes  of  symbols  which  de- 
note quantities  : 

1st.   The  ten  characters,  called  figures : 
2d.   The  straight  line  and  curve  ;   and 
3d.   The  letters  of  the  alphabet. 


ARITHMETIC. 

§  56.  Arithmetic  is  that  branch  of  Mathematics  in 
which  the  quantities  considered  are  represented  by 
figures.  Hence,  it  is  the  science  of  Numbers,  when 
those  numbers  are  represented  by  figures. 


GEOMETRY. 

§  57.  Geometry  is  that  branch  of  Mathematics 
which  treats  of  the  properties  and  relations  of  Lines, 
Surfaces,  Yolumes,  and  Angles,  when  represented  by 
the  pictorial  language  of  which  the  straight  line  and 
curve  are  the  elements.  Hence,  Geometry  is  the 
science  of  space,  when  space  is  represented  by  means 
of  the  straight  line  and  curve. 


NATURE    AND    LANGUAGE.  47 


ALGEBRA. 

§  58.  Algebra  is  that  branch  of  Mathematics  in 
which  the  quantities  considered  are  numerical,  and  de- 
noted by  letters  ;  and  the  operations  to  be  performed 
on  them,  are  indicated  by  signs. 

ANALYSIS. 

§  59.  Analysis  is  a  general  term  embracing  all  the 
operations  which  can  be  performed  on  quantities  when 
represented  by  letters.  In  this  branch  of  mathematics, 
all  the  quantities  considered,  whether  of  number  or 
space,  are  represented  by  letters  of  the  alphabet,  and 
the  operations  to  be  performed  on  them  are  indicated 
by  a  few  arbitrary  signs. 

Analysis,  in  its  simplest  form,  takes  the  name  of 
Algebra ;  Analytical  Geometry,  the  Differential  and 
Integral  Calculus,  extended  to  include  the  Theory  of 
Variations,  are  its  higher  and  most  advanced  branches. 

THE  BRANCHES   OF   MATHEMATICS   DETERMINED   BY 
LANGUAGE. 

§  60.  We  have  seen  that  when  quantities  are  de- 
noted by  figures,  the  operations  belong  to  Arithmetic  : 
when  represented  by  lines,  to  Geometry ;  and  when 
denoted  by  letters,  to  Algebra.     Hence,  the  divisions 


48  MATHEMETICS. 

of  Mathematical  Science  into  the  subjects  of  Arith- 
metic, Geometry,  and  Algebra,  arises  entirely  from  the 
language  employed.  The  methods  of  reasoning,  and 
the  laws  governing  the  operations,  are  the  same  in  all. 

SIGNS     OF    OPERATION. 

§  61.  The  signs  of  operation  are  well  known.  .  They 
are  named  here,  merely  to  preserve  the  symmetry 
in  treating  the  subject.     They  are, 

+,  called  plus,  and  denoting  addition. 

— ,  called  minus,  and  denoting  subtraction. 

x,  called,  the  sign  of  multiplication. 

<— ,  called,  the  sign  of  division. 

=,  called,  the  sign  of  equality. 

v/,  called,  the  radical  sign,  or  the  sign  of  square 
root. 

REMARKS    ON    MATHEMATICAL    LANGUAGE. 

§  62.  The  alphabet  of  the  Arithmetical  language,  as 
already  shown,  contains  ten  characters,  called  figures, 
each  of  which  has  a  name,  and  when  standing  by  itself 
denotes  as  many  things  as  the  name  indicates.  There 
are  but  three  combinations  of  these  characters— the 
first  is  formed  by  writing  them  in  rows — the  second 
by  writing  some  of  them  over  or  under  others — and 
the  third,  by  using  of  the  decimal  point. 


NATURE     AND     LANGUAGE.  49 

This  language,  having  ten  elements  and  three  com- 
binations, is  more  simple,  more  minute,  and  more 
exact  than  any  other  known  form  of  expressing  our 
thoughts.  It  records  all  the  daily  transactions  of  the 
world,  involving  number  and  quantity.  The  yearly 
income — the  accumulations  of  property — the  balance 
sheets  of  mercantile  enterprise  are  all  expressed  in 
numbers,  and  may  be  written  in  figures.  These  ten 
little  characters  are  not  only  the  sleepless  sentinels 
of  trade  and  commerce,  but  they  also  make  known 
all  the  practical  results  of  scientific  labor. 

§463.  The  language  of  Geometry  is  pictorial,  and  has 
but  two  elements,  the  straight  line  and  curve.  The 
combinations  of  these  simple  elements  give  every 
form  and  variety  of  the  geometrical  language.  Dis- 
tance, surface,  volume,  and  angle,  are  names  denoting 
portions  of  space. 

Under  these  four  names  every  part  of  space,  in 
form,  extent,  and  dimension,  is  represented  to  the 
mind  by  means  of  the  straight  line  and  curve.  This 
language  is  both  simple  and  comprehensive.  The 
shortest  distance — the  curve  of  grace  and  beauty — 
the  smooth  surface  and  the  rugged  boundary,  are 
alike  amenable  to  its  laws.  It  presents  to  the  mind, 
through  the  eye,  the  forms  and  relative  magnitudes 
of  all  the  heavenly  bodies,  and,  also,  of  the  most 
minute  and  delicate  objects  that  are  revealed   by  the 

3 


50  MATHEMATICS. 

microscope.  It  is  the  connecting  link  between  theo- 
retical and  practical  knowledge  in  the  mechanic  arts, 
and  the  only  language  in  which  science  speaks  to 
labor.  All  the  works  of  Architecture,  Sculpture,  and 
Painting,  are  but  images  of  the  imagination  until 
they  assume  the  geometrical  forms. 

§  64.  The  language  of  analysis  is  more  comprehen- 
sive than  that  of  figures,  or  the  pictorial  language 
of  Geometry ;  indeed,  it  embraces  them  both.  Its 
elements  are  the  leading  and  final  letters  of  the  al- 
phabet, and  a  few  arbitrary  signs.  The  combinations 
of  these  elements  are  few  in  number  and  simple.. in 
form ;  and  from  these  humble  sources  are  derived 
the  fruitful  language  of  analytical  science. 

This  language  is  minute,  suggestive,  certain,  gen- 
eral, and  comprehensive.  It  will  express  every  prop- 
erty and  relation  of  number — every  form  which  the 
imagination  has  given  to  space — every  moment  of 
time  which  has  elapsed  since  hours  began  to  be  num- 
bered— and  every  motion  which  has  taken  place  since 
matter  began  to  move.  One  or  the  other  of  these 
three  forms  of  mathematical  language  is  in  daily  use 
in  every  part  of  the  world,  and  especially  so  in  every 
place  where  science  is  employed  to  guide  the  hand 
of  labor — to  investigate  the  laws  of  matter — or  to 
enlarge  the  boundaries  of  knowledge. 


NATURE     AND     LANGUAGE.  51 

§  65.  We  have  thus  completed  a  very  brief  and 
genera]  'analytical  view  of  Mathematical  Science. 
We  have  endeavored  to  point  out  the  character  of 
the  definitions,  and  the  sources  as  well  as  the  nature 
of  the  elementary  and  intuitive  propositions  on  which 
the  science  rests  ;  the  kind  of  reasoning  employed 
in  its  creation,  and  its  divisions  resulting  from  the 
use  of  different  symbols,  and  differences  of  language 
We  shall  now  proceed  to  treat  the  subjects  sepa 
rately. 


SECTION    III. 

CIENCE    OF   NUMBERS 


FIRST     NOTIONS     OF     NUMBERS. 

§  66.  There  is  but  a  single  elementary  idea  in  the 
science  of  numbers:  it  is  the  idea  of  the  unit  one. 
There  is  but  one  way  of  impressing  this  idea  on  the 
mind.  It  is  by  presenting  to  the  senses  a  single 
object ;   as,  one  apple,  one  peach,  one  pear,  &c. 

§  67.  There  are  three  signs  by  means  of  which  the 
idea  of  one  is  expressed  and  communicated.  They 
are, 

1st.   The  word  one. 
2d.   The  Roman  character  I. 
3d.   The  figure  1. 

The  idea  of  one,  may  also  be  communicated  through 
the  ear,  by  the  sound,  one. 

§  68.  If  one  be  added  to  one,  the  idea  thus  arising 
is  different  from  the  idea  of  one,  and  is  complex. 
This  new  idea  has  also  three  signs ;  viz.  two,  II.,  and  2. 


INTEGRAL     NUMBERS.  53 

If  one   be    again    added,  that  is,  added  to   two,   the 
new   idea   has   likewise  three   signs ;   viz.  three,  III., 
and    3.      The   ideas   thus   arising,   and   similar   ideas, 
are  called  numbers ;  hence, 
A  number  is  a  unit,  or  a  collection  of  units 

IDEAS     OF     NUMBERS     GENERALIZED. 

§  69.  If  we  begin  with  the  idea  of  the  number  one, 
and  then  add  it  to  one,  making  two ;  and  then  add 
it  to  two,  making  three;  and  then  to  three,  making 
four;  and  then  to  four,  making  five,  and  so  on;  it 
is  plain  that  we  shall  form  a  series  of  numbers,  each 
of  which  will  be  greater  by  one  than  that  which 
precedes  it.  Now,  one  or  a  unit,  is  the  base  of  this 
series  of  numbers,  and  each  number  may  be  expressed 
in  three  ways: 

1st.  By  the  words  one,  two,  three,  &c,  of  our 
common  language  ; 

2d.   By  the  Roman  characters ;   and, 
3d.    By  figures. 

§  70.  Since  all  numbers,  whether  integral  or  frac- 
tional, must  come  from,  and  hence  be  connected  with, 
the  unit  one,  it  follows  that  there  is  but  one  purely 
elementary  idea  in.  the  science  of  numbers.  Hence, 
the  idea  of  every  number,  regarded  as  made  up  of 
units  (and  all  numbers  except  one  must  be  so  regarded, 


54  MATHEMATICS 


when  we  analyze  them),  is  necessarily  complex.  For, 
since  the  number  arises  from  the  addition  of  ones, 
the  apprehension  of  it  is  incomplete  until  we  under- 
stand how  these  additions  were  made;  and  therefore, 
a  full  idea  of  any  number  is  necessarily  complex. 

§  71.  But  if  we  regard  a  number  as  an  entirety, 
that  is,  as  an  entire  or  whole  thing,  as  an  entire  two, 
or  three,  or  four,  without  pausing  to  analyze  the 
units  of  which  it  is  made  up,  it  may  then  be  re- 
garded as  a  simple  or  incomplex  idea ;  though,  as 
we  have  seen,  such  idea  may  always  be  traced  to 
that  of  the  unit  one,  which  forms  the  base  of  the 
number. 

UNITY     AND     A     UNIT     DEFINED. 

§  72.  "When  we  name  a  number,  as  twenty  feet, 
two  things  are  necessary  to  its  clear  apprehension. 

1st.  A  distinct  apprehension  of  the  single  thing 
which  forms  the  base  of  the  number;    and, 

2d.  A  distinct  apprehension  of  the  number  of 
times  which  that  thing  is  taken. 

Any  number,  or  thing,  regarded  as  an  entirety,  or 
whole,  is  called,  unity. 

A  single  thing  which  forms  the  base  of  a  number, 
is  called,  a  unit. 

For  example,  one  foot,  one  rod,  or  twenty  rods, 
regarded  as  a  standard  of  measure,  is  called,  unity: 


INTEGRAL     NUMBERS.  55 

but  one  foot  considered  as  one  of  the  twenty  equal 
parts  of  twenty  feet,  is  called  a  unit,  and  is  the  base 
of  the  number  twenty  feet. 

ALPHABET — WORDS — GRAMMAR. 

§  73.  The  term  alphabet,  in  its  most  general  sense, 
denotes  a  set  of  characters  which  form  the  elements 
of  a  written  language. 

When  any  one  of  these  characters,  or  any  combi- 
nation of  them,  is  used  as  the  sign  of  a  distinct 
notion  or  idea,  it  is  called  a  word ;  and  the  naming 
of  the  characters  of  which  the  word  is  composed,  is 
called,  its  spelling. 

Grammar,  as  a  science,  treats  of  the  established 
connection  between  words,  as  the  signs  of  ideas. 

ARITHMETICAL      ALPHABET. 

§  74.  The  arithmetical  alphabet  consists  of  ten 
characters,  called  figures.     They  are, 

Naught,     One,      Two,      Three,      Four,      Five,        Six,      Seven,       Eight,      Nine, 

01234567        89 

and  each  may  be  regarded  as  a  word,  since  it  stands 
for  a  distinct  idea. 

WORDS — SPELLING    AND    READING    IN    ADDITION. 

§  75.  The  idea  of  one,  being  elementary,  the  char- 
acter 1  which  represents  it,   is  also  elementary,  and 


56  MATHEMATICS. 


hence,  cannot  be  spelled  by  the  other  characters  of 
the  Arithmetical  Alphabet.  But  the  idea  which  is 
expressed  by  2  comes  from  the  addition  of  1  and  1 : 
hence,  the  word  represented  by  the  character  2,  may 
be  spelled  by  1  and  1.  Thus,  1  and  1  are  2,  is  the 
arithmetical  spelling  of  the  word  two. 

Three  is  spelled  thus:  1  and  2  are  3';  and  also, 
2  and  1  are  3. 

Four  is  spelled,  1  and  3  are  4;  3  and  1  are  4; 
2   and  2  are  4. 

Five  is  spelled,  1  and  4  are  5  ;  4  and  1  are  5  ; 
2  and  3  are  5 ;   3  and  2  are  5. 

Six  is  spelled,  1  and  5  are  6;  5  and  1  are  6;  2 
and  4  are  6 ;    4  and  2  are  6 ;    3  and  3  are  6. 

§  76.  In  a  similar  manner,  any  number  in  arithmetic 
may  be  spelled ;  and  hence,  we  see  that  the  process 
of  spelling  in  addition  consists  simply,  in  naming  any 
two  elements  which  will  make  up  the  number.  All 
the  numbers  in  addition  are  therefore  spelled  with 
two  syllables.  The  reading  consists  in  naming  only 
the  word  which  expresses  the  final  idea.     Thus, 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1 

1 

1 

1 

1 

1 

1 

1 

1 

,1 

One 

two 

three 

four 

five 

six 

seven 

eight 

nine 

ten. 

We  may  now  read   the  words  which  express  the 
first  hundred  combinations. 


INTEGRAL    NUMBERS. 


57 


READINGS. 

123456789   10 
1111111111 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

7 

7 

7 

7 

7 

7 

7 

7 

7 

7 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

8 

8 

8 

8 

8 

8 

8 

8 

8 

8 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

9 

9 

9 

9 

9 

9 

9 

9 

9 

9 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

10 

3* 


58  MATHEMATICS. 

§  77.  In  this  example,  beginning  at  the 
right  hand,  we  say,  8,  17,  18,  26:  setting 
down  the  6  and  carrying  the  2,  we  say, 
8,  13,  20,  22,  29 :  setting  down  the  9  and 
carrying  the  2,  we  say,  9,  12,  18,  22,  30: 
and  setting  down  the  30,  we  have  the  entire 
sum,  3096.  All  the  examples  in  addition 
may  be  read  in  a  similar  manner. 


§  78.  The  advantages  of  this  method  of  reading, 
over  spelling,  are  very  great. 

1st.  The  mind  acquires  ideas  more  readily  through 
the  eye  than  through  either  of  the  other  senses. 
Hence,  if  the  mind  be  taught  to  apprehend  the  result 
of  a  combination,  by  merely  seeing  its  elements,  the 
process  of  arriving  at  it  is  much  shorter  than  when 
those  elements  are  presented  through  the  instru- 
mentality of  sound.  Thus,  to  see  4  and  4,  and  think 
8,  is  a  very  different  thing  from  saying,  four  and  four 
are  eight. 

2d.  The  mind  operates  with  greater  rapidity  and 
certainty,  the  nearer  it  is  brought  to  the  ideas  which 
it  is  to  apprehend  and  combine.  Therefore,  all  unne- 
cessary w^ords  load  it  and  impede  its  operations. 
Hence,  to  spell  when  wre  can  read,  is  to  fill  the  mind 
with  words  and  sounds,  instead  of  ideas. 

3d.  All  the  operations  of  arithmetic,  beyond  the 
elementary  combinations, ^re  performed  on  paper;  and  . 


INTEGRAL    NUMBERS.  59 

if  rapidly  and  accurately  done,  must  be  done  through 
the  eye  and  by.  reading.  Hence  the  great  importance 
of"  beginning  early  with  a  method  which  must  be 
acquired  before  any  considerable  skill  can  be  attained 
in  the  use  of  figures. 

§  79.  It  must  not  be  supposed  that  the  reading 
can  be  accomplished  until  the  spelling  has  first  been 
learned. 

In  our  common  language,  we  first  learn  the  alpha- 
bet, then  we  pronounce  each  letter  in  a  word,  and 
finally,  we  pronounce  the  word.  "We  should  do  the 
same  in  the  arithmetical  readings. 

WORDS — SPELLING  AND   READING  IN   SUBTRACTION. 

§  80.  The  processes  of  spelling  and  reading  which 
we  have  explained  in  the  addition  of  numbers,  may, 
with  slight  modifications,  be  applied  in  subtraction. 
Thus,  if  we  are  to  subtract  2  from  5,  we  say,  ordi- 
narily, 2  from  5  leaves  3 ;  or,  2  from  5,  three  remains. 
Now,  the *word,  three,  is  suggested  by  the  relation  in 
which  2  and  5  stand  to  each  other,  and  this  word 
may  be  read  at  once.  Hence,  the  reading ',  in  sub- 
traction, is  simply  naming  the  word,  which  expresses 
the  difference  between  the  minuend  and  subtrahend. 
Thus,  we  may  read  each  word  of  the  following  one 
hundred  combinations. 


60 


MATHEMATICS. 


READINGS. 


1 

1 

2 
1 

3 
1 

4 
1 

5 
1 

6 
1 

7 
1 

8 
1 

9 
1 

10 

1 

2 
2 

3 
2 

4 

2 

5 

2 

6 

2 

7 
2 

8 

2 

9 

2 

10 

2 

11 

2 

3 
3 

4 
3 

5 
3 

6 
3 

7 
3 

8 
3 

9 
3 

10 
3 

11 

3 

12 
3 

4 

4 

5 
4 

6 
4 

7 
4 

8 
4 

9 
4 

10 
4 

11 

4 

12 

4 

13 
4 

5 
5 

6 
5 

7 
5 

8 
5 

9 
5 

10 
5 

11 
5 

12 
5 

13 
5 

14 
5 

6 
6 

7 
6 

8 
6 

9 
6 

10 
6 

11 

6 

12 

.6 

13 
6 

14 

6 

15 
6 

7 
7 

8 

7 

9 

7 

10 

7 

11 

7 

12 

7 

13 

7 

14 

7 

15 

7 

16 

7 

8 
8 

9 

8 

10 

8 

11 

8 

12 

8 

13 

8 

14 

8 

15 

8 

16 

8 

17 

8 

9 
9 

10 
9 

11 
9 

12 
9 

13 
9 

14 

9 

15 
9 

16 
9 

17 
9 

18 
9 

10 
10 

11 

10 

12 

10 

13 

10 

14 
10 

15 
10 

16 
10 

17 

10 

18 
10 

19 
10 

INTEGRAL     NUMBERS.  61 

§  81.  It  should  be  remarked,  that  in  subtraction, 
as  well  as  in  addition,  the  spelling  of  the  words  must 
necessarily  precede  their  reading.  The  spelling  con- 
sists in  naming  the  figures  with  which  the  operation 
is,  performed,  the  steps  of  the  operation,  and  the  final 
result.  The  reading  consists  in  naming  the  final  result 
only. 

SPKLLING  AND   READING-  IN   MULTIPLICATION. 

§  82.  Spelling  in  multiplication  is  similar  to  the 
corresponding  process  in  addition  or  subtraction.  It 
is  simply  naming  the  two  elements  which  produce 
the  product;  whilst  the  reading  consists  in  naming 
only  the  word  which  expresses  the  final  result. 

In  multiplying  each  number  from  1  to  10  by  2, 
we  usually  say,  two  times  1  are  2  ;  two  times  2  are  4 ; 
two  times  3  are  6  ;  two  times  4  are  8 ;  two  times  5 
are  10  ;  two  times  6  are  12  ;  two  times  7  are  14 ;  two 
times  8  are  16;  two  times  9  are  18;  two  times  10 
are  20;  two  times  11  are  22;  two  times  12  are  24. 
Whereas,  we  should  merely  read,  and  say,  2,  4,  6,  8, 
10,  12,  14,  16,  18,  20,  22,  24. 

In  a  similar  manner  we  read  the  entire  multipli- 
cation table. 

READINGS. 

12    11    10      987654321 

1 


62 


MATHEMATICS. 


12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

2 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 
3 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

4 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 
5 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

6 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

7 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

8 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 

9 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 
10 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 
11 

12 

11 

10 

9 

8 

7 

6 

5 

4 

3 

2 

1 
12 

INTEGRAL    NUMBERS.  63 


SPELLING  AND   READING   IN   DIVISION. 

§  83.  In  all  the  cases  of  short  division,  the  quotient 
may  be  read  immediately  without  naming  the  process 
by  which  it  is  obtained.  Thus,  in  dividing  the  follow- 
ing numbers  by  2,  we  merely  read  the  words  below. 

2)4        6        8        10        12        16        18        22 

two       three       four  five  six  eight  nine         eleven. 

In  a  similar  manner,  all  the  words,  expressing  the 
results  in  short  division,  may  be  read. 


READINGS. 

2)2    4      6      8    10    12    14    16    18    20    22    24 


3)3  6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4)4  8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5)5  10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6)6   12 

18 

24 

30 

36 

42 

48 

54 

60 

66 

72 

7)7  14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

8)8  16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

9)9  18 

27 

36 

45 

54 

63 

72 

81 

90 

99  108 

64  MATHEMATICS. 


10)10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

120 

11)11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12)12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

UNITS   INCREASING  BY   THE   SCALE   OF   TENS. 

§  84.  The  idea  of  a  particular  number  is  necessarily 
complex ;  for,  the  mind  naturally  asks  : 

1st.  What  is  the  unit  or  base  of  the  number?   and, 
2d.  How  many  times  is  the  unit  or  base  taken  ? 

§  85  A  figure  indicates  how  many  times  a  unit  is 
taken.  Each  of  the  ten  figures,  however  written,  or 
however  placed,  always  expresses  as  many  units  as 
its  name  imports,  and  no  more ;  nor,  does  the  figure 
itself  at  all  indicate  the  kind  of  unit. 

Still,  every  number  expressed  by  one  or  more 
figures,  has  for  its  base  either  the  abstract  unit  one, 
or  a  denominate  unit.  If  a  denominate  unit,  its 
value  or  kind  is  pointed  out  either  by  our  common 
language,  or,  as  we  shall  presently  see,  by  the  place 
where  the  figure  is  written. 

The  number  of  units  which  may  be  expressed  by 
either  of  the  ten  figures,  is  indicated  by  the  name 
of  the  figure.      If  the   figure  stands   alone,   and   the 


INTEGRAL     NUMBERS.  65 

unit  is  not  denominated,  the  base   of  the  number  is 
the  abstract  unit  1. 

§86.  If  we  write  0  on  the  right  of  1,  we  ) 
have J 

which  is  read   one   ten.     Here   1   still  expresses  one, 

but  it  is  one  ten ;  that  is,  a  unit  ten  times  as  great 

as  the  unit  1 ;  and  this  is  called  a  unit  of  the  second 

order. 

Again :  if  we  write  two  O's  on  the  right  of ) 
.      *■     '                                                  *  [    100, 

1,  we  have ) 

which   is    read    one    hundred.      Here    again,   1    still 

expresses    one,    but   it   is   one    hundred  ;    that    is,    a 

unit   ten    times  as  great  as  the  unit  one  ten,   and   a 

hundred   times  as  great  as  the  unit  1. 

§  87.  If  three   l's  are  written  by  the  side  ) 
of  each  other,  thus    ..,....) 
the   ideas,   expressed    in    our    common   language,   are 
these : 

1st.  That  the  1  on  the  right,  will  either  express  a 
single  thing  denominated,  or  the  abstract  unit  one. 

2d.  That  the  1  next  at  the  left,  expresses  1  ten, 
that  is,  a  unit  ten  times  as  great  as  the  first ; 

3d.  That  the  1  still  further  to  the  left,  expresses 
1  hundred-;  that  is,  a  unit  ten  times  as  great  as 
the  second,  and  one  hundred  times  as  great  as  the 
first  /   and  similarly  if  there  are  other  places. 


66  MATHEMATICS. 

"When  figures  are  thus  written  by  the  side  of 
each  other,  the  arithmetical  language  establishes  a 
relation  between  the  units  of  their  places:  that  is, 
the  unit  of  each  place,  as  we  pass  from  the  right 
hand  towards  the  left,  increases  according  to  the 
scale  of  tens.  Therefore,  by  a  law  of  the  arithmet- 
ical language,  the  place  of  a  figure  fixes  its  unit. 

If,  then,  we  write  a  row  of  O's  as  a  scale,  thus : 


3        a 

2        5      « 


00    0,      00     0,      00    0,      000 

the  unit  of  each  place  is  determined,  as  well  as  the 
law  of  change  in  passing  from  one  place  to  another. 
If  then,  it  were  required  to  express  a  given  number 
of  units,  of  any  order,  we  first  select  from  the  arith- 
metical alphabet  the  character  which  designates  the 
number,  and  then  write  it  in  the  place  correspond- 
ing to  the  order.  Thus,  to  express  3  unit  of  the 
seventh  order,  or  three  millions,  we  write 

3000000; 

and  similarly  for  all  numbers. 

§  88.   It  should    be   observed,   that   a   figure  being 
a  character  which  represents  value,  can  have  no  value 


INTEGRAL     NUMBERS.  67 

in  and  of  itself.  The  number  of  things,  which  any 
figure  expresses,  is  determined  by  its  name,  as  given 
in  the  arithmetical  alphabet.  The  kind  of  thing, 
or  unit  of  the  figure,  is  fixed  either  by  naming  it, 
as  in  the  case  of  a  denominate  number,  or  by  the 
place  which  the  figure  occupies,  when  written  by 
the  side  of  or  over  other  figures. 

The  phrase  "  local  value  of  a  figure,"  so  long  in 
use,  is,  therefore,  without  signification  when  applied 
to  a  figure  :  the  term  "  local  value,"  being  appli- 
cable to  the  unit  of  the  place,  and  not  to  the  figure 
which   occupies  the   place. 

§  89.  A  scale  is  a  series  of  numbers  expressing 
the  law  of  relation  between  the  different  units  of 
any  number. 

A  Uniform  Scale  is  one  in  which  the  law  of 
relation  between  the  units,  at  any  step  of  the  scale, 
is  the  same. 

A  Varying  Scale  is  one  in  which  the  law  of  re- 
lation between  the  units,  is  different,  at  different 
steps  of  the  scale. 

The  Units  of  the  Scale,  at  any  step,  are  denoted 
by  the  number  of  units  of  the  lower  denomination 
which  makes  one  unit   of  the  next  higher. 

§  90.  United  States  money  affords  an  example  of  a 
series  of  denominate  units,  increasing  according  to  the 
scale  of  tens:   thns, 


68  MATHEMATICS. 


W      ft      ft      q      S3 
11111 

may  be  read  11  thousand  1  hundred  and  11  mills; 
or,  1111  cents  and  1  mill  ;  or,  111  dimes  1  cent 
and  1  mill  ;  or,  11  dollars  1  dime  1  cent  and  1 
mill  ;  or,  1  eagle  1  dollar  1  dime  1  cent  and  1 
mill.  Thus,  we  may  read  the  number  with  either 
of  its  units  as  a  base,  or  we  may  name  them  all : 
thus,  1  eagle,  1  dollar,  1  dime,  1  cent,  1  mill.  Gen- 
erally, in  United  States  Money,  we  read  in  the  de- 
nominations of  dollars,  cents,  and  mills ;  and  should 
say,  11  dollars  11  cents  and  1  mill. 

§  91.  Examples  in  reading  figures : — 

If  we   have  the  figures 89 

we   may  read  them   by  their  smallest  unit, 

and  say  eighty-nine ;   or,   we  may  say   8  tens   and   9 

units. 

Again,  the  figures 567 

may   be   read   by    the    smallest   unit ;    viz. 

five  hundred   and    sixty-seven ;    or  we   may   say,   56 

tens    and    7   units  ;    or,    5    hundreds    6    tens    and    7 

units. 

Again,  the  number  expressed  by       .      .     74S96 
may    be   read,   seventy-four   thousand   eight 
hundred   and   ninety-six.      Or,  it   may  be  read,  7189 


INTEGRAL     NUMBERS.  69 

tens  and  6  units;  or,  748  hundreds  9  tens  and  6 
units  ;  or,  74  thousands  8  hundreds  9  tens  and  6 
units  ;  or,  7  ten  thousands  4  thousands  8  hundreds 
9  tens  and  6  units ;  and  we  may  read  in  a  similar 
way  all  other  numbers. 

Although  we  should  teach  all  the  correct" readings 
of  a  number,  we  should  not  fail  to  remark  that  it 
is  generally  most  convenient  in  practice  to  read  by 
the  lowest  unit  of  a  number.  Thus,  in  the  nume- 
ration table,  we  read  each  period  by  the  lowest 
unit  of  that  period.     For    example,  in  the  number 

874,967,847,047, 

we  read  874  billions  967  millions  847  thousands  and 
47. 

UNITS   INCREASING    ACCORDING   TO   VARYING   SCALES. 

§  92.  If  we  write  the  well-known  signs  of  the  Eng- 
lish money,  and  place  1  under  each  denomination, 
we  shall  have 

£      s.       d.      f. 
1111 

Now,  the  signs  £  s.  d.  and  f.  fix  the  value  of  the 
unit  1  in  each  denomination  ;  and  they  also  deter- 
mine the  relations  which  subsist  between  the  differ- 


70  MATHEMATICS. 

ent  units.  For  example,  this  simple  language  ex- 
presses these  ideas  : 

1st.  That  the  unit  of  the  right-hand  place  is  1  far- 
thing— of  the  place  next  to  the  left,  1  penny — of  the 
next  place,  1  shilling — of  the  next  place,  1  pound ; 
and 

2.  That  4  units  of  the  lowest  denomination  make 
one  unit  of  the  next  higher;  12  of  the  second,  one  of 
the  third ;  and  20  of  the  third,  one  of  the  fourth. 

§  93.  If  we  take  the  denominate  numbers  of  the 
Avoirdupois  weight,  we  have 

Ton    cwt.     qr.      lb.'     oz.       dr. 
11111       1; 

in  which  the  units  increase  in  the  following  manner: 
viz.  the  second  unit,  counting  from  the  right,  is  six- 
teen times  as  great  as  the  first;  the  third,  sixteen 
times  as  great  as  the  second ;  the  fourth,  twenty- 
five  times  as  great  as  the  third ;  the  fifth,  four  times 
as  great  as  the  fourth  :  and  the  sixth,  twenty  times 
as  great  as  the  fifth.  The  scale,  therefore,  for  this 
class  of  denominate  numbers  varies  according  to  the 
above  laws. 

If  we  take  any  other  class  of  denominate  numbers, 
as  the  Troy  weight,  or  any  of  the  systems  of  meas- 
ures, we  shall  have  different  scales  for  the  formation 
of  the  different  units.     But  in  all  the  formations,  we 


INTEGRAL    NUMBERS.  71 

recognize  the  application  of  the  same  general  prin- 
ciples. 

There  are,  therefore,  two  general  methods  of  forming 
the  different  systems  of  integral  numbers  from  the 
unit  one.  The  first  consists  in  preserving  a  constant 
law  of  relation  between  the  different  unities ;  viz.  that 
their  values  shall  change  according  to  the  scale  of 
tens.     This  gives  the  system  of  common  numbers. 

The  second  method  consists  in  the  application  of 
known,  though  varying  laws  of  change  in  the  uni- 
ties. These  changes  in  the  unities  produce  the  entire 
system  of  denominate  numbers,  each  class  of  which 
has  its  appropriate  scale;  and  the  changes  among  the 
units,  of  the  same  class,  are  indicated  at  the  different 
steps  of  the  scale. 

INTEGRAL     UNITS     OF     ARITHMETIC. 

§  94.  There  are  eight  classes  of  units  in  Arithmetic : 

1st.  Abstract  Units. 

2d.  Units  of  Currency. 

3d.  Units  of  Weight. 

4th.  Units  of  Time. 

5th.  Units  of  Length. 

6th.  Units  of  Surface. 

7th.  Units  of  Yolume. 

8th.  Units  of  Angular  Measure. 
First  among:  the  Units  of  arithmetic  stands  the  ab- 


72  MATHEMATICS 


stract  unit  1.  This  is  the  base  of  all  abstract  num- 
bers, and  becomes  the  base,  also,  of  all  denominate 
numbers,  by  merely  naming,  in  succession,  the  par- 
ticular things  to  which  it  is  applied. 

It  is  also  the  base  of  all  fractions.  Merely  as  the 
unit  1,  it  is  a  whole  which  may  be  divided  according 
to  any  law,  forming  every  variety  of  fraction  ;  and  if 
we  apply  it  to  a  particular  thing,  the  fraction  be- 
comes denominate,  and  we  have  expressions  for  all 
conceivable  parts  of  that  thing. 

§  95.  It  has  been  remarked  that  we  can  form  no 
distinct  apprehension  of  the  number,  until  we  have 
a  clear  notion  of  its  unit,  and  the  number  of  times 
the  unit  is  taken.  The  unit  is  the  great  base.  The 
utmost  care,  therefore,  should  be  taken  to  impress  on 
the  minds  of  learners,  a  clear  and  distinct  idea  of 
the  actual  value  of  the  unit  of  every  number  with 
which  they  have  to  do.  If  it  be  a  number  expressing 
currency,  one  or  more  of  the  coins  should  be  ex- 
hibited, and  the  value  dwelt  upon  ;  after  which,  dis- 
tinct notions  of  the  other  units  can  be  acquired  by 
comparison. 

If  the  number  be  one  of  weight,  some  unit  should 
be  exhibited,  as  one  pound,  or  one  ounce,  and  an 
idea  of  its  weight  acquired  by  actually  lifting  it. 
This  is  the  only  way  in  which  we  can  learn  the  true 
signification  of  the  terms. 


INTEGRAL    NUMBERS.  73 

If  j;he  number  be  one  of  measure,  either  linear, 
superficial,  of  volumes  or  angles,  its  unit  should  also 
be  exhibited,  and  the  signification  of  the  term  express- 
ing it,  learned  in  -the  only  way  in  which  it  can  be 
learned,  through  the  senses,  and  by  the  aid  of  a  sensi- 
ble object. 

ABSTRACT     UNITS. 

§  96.  Abstract  units,  are  those  which  denote 
single  things,  in  which  the  base  is  the  abstract  num- 
ber one.  They  are  mere  expressions  of  number,  and 
may  be  applied  to  all  quantities  by  assigning  the 
proper  denominate  unit. 


ORDERS     OF    ABSTRACT    UNITS. 

§  97.  The  abstract  units,  are,  one,  one  ten,  one  hun- 
dred, one  thousand,  one  ten  thousand,  &c,  &c.  They 
are  called  units  of  different  orders.  Thus,  one  is  a 
unit  of  the  first  order;  ten  of  the  second;  one  hun- 
dred of  the  third;  one  thousand  of  the  fourth,  &c. 


UNITS     OF    CURRENCY. 

§  98.  The  Units  of  United  States  currency,  are  1 
mill,  1  cent,  1  dime,  1  dollar,  and  1  eagle.  The  law  of 
change,  in  passing  from  one  unit  to  another,  is  ac- 
cording to  the  scale  of  tens.    Hence,  this  system  of 

i 


74  MATHEMATICS, 


numbers  may  be  treated,  in  all  respects,  as  abstract 
numbers;  and  indeed  they  are  such,  with  the  single 
exception  that  their  units  have  different  names. 

They  are  generally  read  in  the  units  of  dollars, 
cents,  and  mills — a  period  being  placed  after  the  figure 
denoting  dollars.     Thus, 

$  864.849 

is  read,  eight  hundred  and  sixty-four  dollars,  eighty- 
four  cents  and  nine  mills ;  and  if  there  were  a  figure 
after  the  9,  it  would  be  read  in  decimals  of  the  mill. 
The  number  may,  however,  be  read  in  any  other 
unit ;  as,  864849  mills ;  or,  86484  cents  and  9  mills ; 
or,  8648  dimes,  4  cents,  and  9  mills;  or,  86  eagles, 
4  dollars,  84  cents,  and  9  mills ;  and  there  are  yet 
several  other  readings. 


ENGLISH     CURRENCY. 

§  99.  The  units  of  English,  or  Sterling  Money,  are 
1  farthing,  1  penny,  1  shilling,  and  1  pound. 

The  scale  of  this  class  of  numbers  is  a  varying 
scale.  Its  steps,  in  passing  from  the  unit  of  the 
lowest  denomination  to  the  highest,  are  four,  twelve, 
and  twenty.  For,  four  farthings  make  one  penny, 
twelve  pence  one  shilling,  and  twenty  shillings  one 
pound. 


INTEGKAL     KUMBEES.  75 


FRENCH  CURRENCY. 

§  100.  The  units  of  the  French.  Currency  are  the 
Franc,  which  forms  the  base,  and  the  Napoleon, 
equal  to  twenty  francs. 

UNITS     OF    WEIGHT. 

§  101.  There  are  three  kinds  of  Weight  in  general 
use :  Avoirdupois  Weight,  Troy  Weight,  and  Apothe- 
caries' Weight. 

AVOIRDUPOIS     WEIGHT. 

§  102.  The  units  of  the  Avoirdupois  Weight  are, 
1  dram,  1  ounce,  1  pound,  1  quarter,  1  hundred- weight, 
and  1  ton. 

The  scale  of  this  class  of  numbers  is  a  varying 
scale.  Its  steps,  in  passing  from  the  unit  of  the 
lowest  denomination  to  the  highest,  are  sixteen,  six- 
teen, twenty-five,  four,  and  twenty.  For,  sixteen  drams 
make  one  ounce,  sixteen  ounces  one  pound,  twenty- 
five  pounds  one  quarter,  four  quarters  one  hundred, 
and  twenty  hundreds  one  ton. 

TROY     WEIGHT. 

§  103.  The  units  of  the  Troy  Weight  are,  1  grain, 
1  pennyweight,  1  ounce,  and  1  pound. 


76  MATHEMATICS 


The  scale  is  a  varying  scale;  and  its  steps,  in 
passing  from  the  unit  of  the  lowest  denomination  to 
the  highest,  are  twenty-four,  twenty,  and  twelve. 


§  104.  The  units  of  this  weight  are,  1  grain,  1 
scruple,  1  dram,  1  ounce,  and  1  pound. 

The  scale  is  a  varying  scale.  Its  steps,  in  passing 
from  the  unit  of  the  lowest  denomination  to  the 
highest,  are  twenty,  three,  eight,  and  twelve. 

TIME. 

§  105.  The  units  of  Time  are,  1  second,  1  minute, 
1  hour,  1  day,  1  week,  1  month,  1  year,  and  1  century. 
The  steps  of  the  scale,  in  passing  from  units  of  the 
lowest  denomination  to  the  highest,  are  sixty,  sixt}r, 
twenty-four,  seven,  four,  twelve,  and  one  hundred. 

UNITS     OF     LENGTH. 

§  106.  The  unit  of  Length  is  used  for  measuring 
lines,  either  straight  or  curved.  It  is  a  straight  line 
of  a  given  length,  and  is  called  the  standard  or  base, 
of  the  measurement. 

The  units  of  length,  generally  used  as  standards, 
are  1  inch,  1  foot,  1  yard,  1  rod,  1  furlong,  and  1  mile. 


INTEGRAL    NUMBERS, 


77 


The  number  of  times  which  the  unit,  used  as  a 
standard,  is  taken,  considered  in  connection  with,  its 
value,  gives  the  idea  of  the  length  of  the  line 
measured. 


1  square  foot. 


1  yard. 


UNITS    OF     SURFACE. 

§  107.  Units  of  surface  are  used  for  the  measure- 
ment of  the  area  or  contents  of  whatever  has  the  two 
dimensions  of  length  and  breadth. 
The  unit  of  surface  is  a  square  de- 
scribed on  the  unit  of  length  as  a 
side.  Thus,  if  the  unit  of  length  be 
1  foot,  the  corresponding  unit  of 
surface  will  be  1  square  foot;  that  is,  a  square  con- 
structed on  1  foot  of  length,  as  a  side. 

If  the  linear  unit  be  1  yard,  the 
corresponding  unit  of  surface  will  be 
1  square  yard.  It  will  be  seen  from 
the  figure,  that,  although  the  linear 
yard  contains  the  linear  foot  but  three 
times,  the  square  yard  contains  the 
square  foot  nine  times.  The  square 
rod  or  square  mile  may  also  be  used  as  the  unit  of 
surface. 

The  number  of  times  which  a  surface  contains  its 
unit  of  measure,  is  its  area  or  contents ;  and  this 
number,  taken  in  connection  with  the  value  of  the 
unit,  gives  the  idea  of  its  extent. 


78  MATHEMATICS. 

— — — — ———_________ i>  ■ 

Besides  the  units  of  surface  already  considered, 
t&ere  is  another  kind,  called, 

DUODECIMAL     UNITS. 

§  108.  The  duodecimal  units  are  generally  used  in 
board  measure,  though  they  may  be  used  in  all 
superficial  measurements,  and  also  in  volumes. 

The  square  foot  is  the  base  of  this,  class  of  units, 
and  the  others  are  deduced  from  it,  by  a  descending 
scale  of  twelve. 

§  109.  It  is  proved  in  Geometry,  that  if  the  number 
of  linear  units  in  the  base  of  a  rectangle  be  multiplied 
by  the  number  of  linear  units  in  its  height,  the 
numerical  value  of  the  product  will  be  equal  to  the 
number  of  superficial  units  in  the  figure. 

Knowing  this  fact,  wTe  often  express  it  by  saying, 
that  "  feet  multiplied  by  feet  give  square  feet,"  and 
"  yards  multiplied  by  yards  give  square  yards."  But 
as  feet  cannot  be  taken  feet  times,  nor  yards  yard 
times,  this  language,  rightly  understood,  is  but  a 
concise  form  of  expression  for  the  principle  stated 
above. 

"With  this  understanding  of  the  language,  we  say, 
that  1  foot  in  length  multiplied  by  1  foot  in  height, 
gives  a  square  foot ;  and  4  feet  in  length  multiplied  by 
3  feet  in  height,  gives  12  square  feet. 


INTEGRAL    NUMBERS.  79 

§  110.  If  now,  1  foot  in  length 
be  multiplied  by  1  inch  =  T\  of  a 
foot  in  height,  the  product  will  be 
one-twelfth  of  a  square  foot ;  that  is, 
one-twelfth  of  the  first  unit :  if  it  be 
multiplied  by  3  inches,  the  pro- 
duct will  be  three-twelfths  of  a  square  foot;  and 
similarly,  for  a  multiplier  of  any  number  of  inches. 

If,  now,  we  multiply  1  inch  by  1  inch,  the  product 
may  be  represented  by  1  square  inch:  that  is,  by  one- 
twelfth  of  the  last  unit.  Hence,  the  units  of  this 
measure  decrease  according  to  the  scale  of  12.  The 
units  are, 

1st.  Square  feet — arising  from  multiplying  feet  by 
feet: 

2d.  Twelfths  of  square  feet — arising  from  multiply- 
ing the  denomination  of  feet  by  the  denomination  of 
inches : 

3d.  Twelfths  of  twelfths — arising  from  multiplying 
inches  by  inches. 

The  same  remarks  apply  to  the  smaller  divisions  of 
the  foot,  according  to  the  scale  of  twelve. 

The  difficulty  of  computing,  in  this  measure,  arises 
from  the  changes  in  the  units. 

UNITS     OF    VOLUME. 

§  111.  Volume  has  been  denned  in  Geometry  to  be 
a  limited  portion  of  space.     It  has  already  been  stated, 


80  MATHEMATICS. 

that  if  length  be  multiplied  by  breadth,  the  product 
may  be  represented  by  units  of  surface.  It  is  also 
proved,  in  Geometry,  that  if  the  length,  breadth,  and 
height  of  any  regular  figure,  of  a  square  form,  be 
multiplied  together,  the  product  may  be  represented 
by  units  of  volume  whose  number  is  equal  to  this 
product.  Each  unit  of  volume  is  a  cube  constructed 
on  the  linear  unit  as  an  edge.  Thus,  if  the  linear  unit 
be  1  foot,  the  unit  of  volume  will  be  1  cubic  foot; 
that  is,  a  cube  constructed  on  1  foot  as  an  edge  ;  and 
if  it  be  1  yard,  the  unit  will  be  1  cubic  yard. 

The  three  units,  viz.  the  unit  of  length,  the  unit  of 
surface,  and  the  unit  of  volume,  are  essentially  dif- 
ferent in  kind.  The  first  is  a  line  of  a  known  length ; 
the  second,  a  square  of  a  known  side;  and  the  third,  a 
volume,  called  a  cube,  of  a  known  base  and  height. 
These  are  the  units  used  in  all  kinds  of  measure- 
ments of  solids,  excepting  only  the  ^duodecimal  system, 
which  has  already  been  explained. 

§  112.  A  volume  filled  with  matter,  is  called  a  solid 
or  body  /  and  for  this,  the  unit  of  measure  is  the  cube, 
except  in  the  case  of  wood,  where  it  is  a  cord  of 
128  cubic  feet. 

But  a  given  volume  filled  with  a  liquid,  takes  the 
name  of  a  gallon  ;  and  a  larger  volume  filled  with 
grain,  takes  the  name  of  a  bushel.  Hence,  the  Liquid 
and  Dry  measures,  are  measures  of  volume,  and  differ 


INTEGRAL     NUMBERS.  81 

only  from  the  measurement  of  solidity,  in   the  form 
and  value  of  the  unit  of  measure. 

LIQUID    MEASURE. 

§  113.  The  units  of  Liquid  Measure  are,  1  gill,  1 
pint,  1  quart,  1  gallon,  1  barrel,  1  hogshead,  1  pipe,  1 
tnn.  The  scale  is  a  varying  scale.  Its  steps,  in  pass- 
ing from  the  unit  of  the  lowest  denomination,  are,  four, 
two,  four,  thirty-one  and  a  half,  sixty-three,  two,  and 
two. 

DRY     MEASURE. 

§  114.  The  units  of  this  measure  are,  1  pint,  1 
quart,  1  peck,  1  bushel,  and  1  chaldron.  The  steps 
of  the  scale,  in  passing  from  units  of  the  lowest  de- 
nomination, are,  two,  eight,  four,  and  thirty -six. 

ANGULAR    MEASURE. 

§  115.  The  units  of  this  measure  are,  1  second, 
1  minute,  1  degree,  1  sign,  1  circle.  The  steps  of  the 
scale,  in  passing  from  units  of  the  lowest  denomina- 
tion to  those  of  the  higher,  are  sixty,  sixty,  thirty,  and 
twelve. 

A* 


82  MATHEMATICS. 


ADVANTAGES    OF   THE    SYSTEM    OF    UNITIES. 

§  116.  It   may  well  be   asked,  if  the   method  here 

adopted,   of   presenting  the   elementary   principles   of 

/arithmetic,   has   any   advantages    over    those  now   in 

general  use.     It  is  supposed  to  possess  the  following : 

1st.  The  system  of  unities  teaches  an  exact  analy- 
sis of  all  numbers,  and  unfolds  to  the  mind  the 
different  ways  in  which  they  are  formed  from  the 
unit  one,  as  a  base. 

2d.  Such  an  analysis  enables  the  mind  to  form 
a  definite  and  distinct  idea  of  every  number,  by 
pointing  out  the  relation  between  it  and  the  unit 
from   which  it  was  derived. 

3d.  By  presenting  constantly  to  the  mind  the  idea 
of  the  unit  one,  as  the  base  of  all  numbers,  the 
mind  is  insensibly  led  to  compare  this  unit  with  all 
the  numbers  which  flow  from  it,  and  then  it  can  the 
more  easily  compare  those  numbers  with   each  other. 

4th.  It  affords  a  more  satisfactory  analysis  and  a 
better  understanding  of  the  four  ground-rules,  and 
indeed  of  all  the  operations  of  arithmetic,  than  any 
other  method  of  presenting  the  subject. 


FOUR     GROUND-RULES. 

§  117.    Let   us    take    the    two    following   examples 
in   Addition,  the   one   in    abstract   and   the   other  in 


INTEGRAL  NUMBERS.                          83 

denominate    numbers,    and  then    analyze  the  process 
of  finding  the  sum  in  each. 

SIMPLE    NUMBERS.  DENOMINATE  NUMBERS. 

874198  cwt.  qr.     lb.  oz.      dr. 

86984  3     3     24  15     14 

3641  6     3     23  14       8 


914823  10     3     23     14       6 


In  both  examples  we  begin  by  adding  the  units 
of  the  lowest  denomination,  and  then,  we  divide 
their  sum  by  so  many  as  make  one  of  the  denom- 
ination next  higher.  We  then  set  down  the  re- 
mainder, and  add  the  quotient  to  the  units  of  that 
denomination.  Having  done  this,  we  apply  a  sim- 
ilar process  to  all  the  other  denominations — the 
principle  being  precisely  the  same  in  both  examples. 
We  see,  in  these  examples,  an  illustration  of  a 
general  principle  of  addition,  viz.  that  units  of  the 
same  hind  are  always  added  together. 

%  118.  Let  us  take  two  similar  examples  in  Sub- 
traction. 


SIMPLE    NTJMBEBS.  DENOMINATE    NUMBERS. 

8403  £       ,.       d.  far. 

3298  6       9       7    2 

5105  .           3     10      8    4 


2     18     10    2 


84:  MATHEMATICS. 

In  both  examples  we  begin  with  the  units  of  the 
lowest  denomination,  and  as  the  number  in  the 
subtrahend  is  greater  than  in  the  place  directly 
above,  we  suppose  so  many  to  be  added  in  the 
minuend  as  make  one  unit  of  the  next  higher  de- 
nomination. We  then  make  the  subtraction,  and 
add  1  to  the  units  of  the  subtrahend  next  higher, 
and  proceed  in  a  similar  manner,  through  all  the 
denominations.  It  is  plain  that  the  principle  em- 
ployed is  the  same  in  both  examples.  Also,  that 
units  of  any  denomination  in  the  subtrahend  are 
taken  from  those  of  the  same  denomination  in  the 
minuend. 


§  119.  Let   us   now  take   similar   examples   in  Mul- 
tiplication. 


SIMPLE    NUMBERS. 

DENOMINATE    NUMBERS. 

87464 

lb      I      3     3     jr. 

5 

9     7     6     2     15 

4.37220 

5 

48     3     2     1     15 


In  these  examples  we  see,  that  we  multiply,  in 
succession,  each  order  of  units  in  the  multiplicand 
by  the  multiplier,  and  that  we  carry  from  one 
product  to  another,  one  for  every  so  many  as  make 
one    unit    of    the    next    higher    denomination.      The 


INTEGRAL     NUMBERS.  85 

principle   of   the   process    is    therefore    the    same    in 
both   examples. 

LOGICAL     FORM     FOR     ADDITION. 

§  120.  Def.  A  number  which  contains  as  many 
units  as  all  the  numbers  added,  is  called  their  sum. 

With  this  definition,  let  it  be  required  to  prove 
tli at  8  is  the  sum  of  3  and  5. 

The  Syllogistic  form  is : 

A  number  which  contains  as  many  units  as  all  the 
numbers  added,  is  called  their  sum.     (Major  Premise.) 

Eight  contains  as  many  units  as  there  are  in  3 
and  5;  (because  3  counted  on  to  5  make  8).  (Minor 
Premise.) 

Therefore,  8  is  the  sum  of  3  and  5.  This  proof, 
logically,  is  perfect,  because  it  brings  the  result  of 
the  operation  performed  on  3  and  5,  under  the 
term,    Sum. 

LOGICAL     FORM     FOR     SUBTRACTION. 

§  121.  Def.  A  number  which  added  to  the  less  of 
two  numbers  will  give  the  greater,  is  called  their 
Difference. 

What  is  the  difference  between  7  and  10? 

The  syllogistic  form  is : 

A  number  wThich  added  to  the  less  of  two  num- 
bers will  give  the  greater,  is  called  their  difference; 
3  added  to  7  gives  10  ; 


86  MATHEMATICS. 

Therefore,  3  is  the  difference  between  7  and  10. 
Here,  again,  the  logical  form  merely  brings  the  re- 
sult of  the  operation  under  the  definition. 

§  122.  In  Multiplication,  if  we  define  the  operation 
to  be,  the  process  of  taking  the  multiplicand  as  many 
times  as  there  are  units  in  the  multiplier,  we  prove  the 
operation  by  showing  that  the  result  fulfils  this  con- 
dition. 

§  123.  So,  in  Division,  if  we  define  the  quotient  to 
be  such  a  number  as  multiplied  by  the  divisor  will 
produce  the  dividend ;  we  prove  the  operation  to 
be  correct,  when  we  show  that  the  number  found  is 
such  a  multiplier. 

Every  proof  in  the  entire  range  of  mathematical 
science,  consists  in  'bringing  the  thing  to  he  proved, 
under  a  definition,  an  axiom,  or  a  proposition  pre- 
viously established. 


METRIC    SYSTEM. 

§  124.  Every  system  of  Weights  and  Measures  must 
have  an  invariable  unit  for  its  base — and  every  other 
unit  of  the  entire  system  should  be  derived  from  it, 
according  to  a  fixed  law. 

The  French  Government,  in  order  to  obtain  an 
invariable  unit,  measured   a  degree  of  the   arc   of  a 


METRIC     SYSTEM.  87 

meridian  on  the  Earth's  surface;  and  from  this  com- 
puted the  length  of  the  meridional  arc  from  the  equator 
to  the  pole.  This  length  they  divided  into  ten  million 
equal  parts,  and  then  took  one  of  these  parts  for  the 
unit  of  length,  and  called  it  a  Meter.  The  length 
of  this  meter  is  equal  to  1  yard,  3  inches  and  37  hun- 
dredths of  an  inch,  very  nearly.  Thus  they  obtained 
the  length  of  the  unit  which  is  the  base  of  the  Metric 
System  of  Weights  and  Measures. 

The  next  step  was  to  fix  the  law  by  which  the  other 
units  of  the  system  should  be  obtained  from  the  base. 
As  the  scale  of  tens  is  the  simplest  law  by  which  we 
can  pass  from  one  unit  to  another,  that  scale  was 
adopted,  and  the  larger  units  are  formed  by  multiply- 
ing the  base  continually  by  10,  and  the  smaller,  by 
dividing  it  continually  by  10. 

NAMING. 

§  125.  The  names,  in  the  ascending  scale,  are  formed 
by  prefixing  to  the  base,  Meter,  the  words  Deca  (ten), 
Hecto  (one  hundred),  Kilo  (one  thousand),  Myria  (ten 
thousand),  from  the  Greek  numerals;  and  in  the 
descending  scale,  by  prefixing  Deci  (tenth),  centi 
(hundredth),  Milli  (thousandth),  from  the  Latin  numerals. 
Hence,  the  name  of  a  unit  indicates  whether  it  is 
greater  or  less  than  the  standard,  and  also,  how  many 
times. 


88  MATHE  M  A  TICS 


MEASUREMENT     OF     LENGTH 

Hence,  for  the  measurement  of  length,  we  have 


Ascend 

tfn^  #mte. 

•fca 

Descending  Scale. 

s 

Eh 

N 

.  CD 

s 

0) 

§3 

a 

c3 
■| 

o 
o 

a 

o 

1 

O 

■4-> 

Q 

a 

o 

Centimeter 

Millimeter 

< 

a 

a 

H 

A 

a 

A 

in  which  the  increase  and  decrease,   from  the  base, 
takes  place  according  to  the  scale  of  tens. 

SQUARE    MEASURE. 

§  126.  The  unit  of  measure  for  surfaces  is  a  square 
described  on  a  line  10  meters  in  length,  and  is  called 
an  Ark.  Hence,  the  unit  of  surface  is  equal  to  100 
square  meters.  It  is  also  equal  to  4  perches,  nearly. 
There  are  but  two  other  units  used  in  the  French 
system — the  Centare,  which  is  one  hundredth  of  the 
Are,  and  the  Hectare,  which  is  one  hundred  times 
the  Are. 

MEASURES    OF    VOLUME. 

§  127.  The  unit  for  the  measurement  of  volumes  is 
the  cube  constructed  on  the  decimeter,  as  an  edge.  It 
is  called  a  Leter  ;  and  is  equal  to  61  cubic  inches,  very 
nearly.     All  the  other  denominations  are  derived  from 


METRICS  Y  STEM.  89 

the  base,  in  the  same  manner  as  in  the  measurement 
of  length.     Thus,  we  have 


Ascending  Scale 

i— i 
O 

P 

J 

M 
I 

>3 

Descending  Scale. 

Kiloleter 
Hectoleter 

tit 

<v 

<D 

A 

Centileter 
Millileter 

DRY    MEASURE. 


§  128.  The  Leter  is  also  the  unit  for  dry  measure. 
It  is  a  little  less  than  1  quart  of  the  Winchester  bushel. 


LIQUID     MEASURE. 

§  129.  The  unit  of  this  measure  is  also  the  Leter. 
It  contains  a  little  more  than  3  pints  of  the  wine 
gallon. 

WEIGHTS. 

§  130.  The  unit  of  weight  is  the  weight  of  a  cubic 
centimeter  of  pure  rain-water,  weighed  in  vacuum. 
Hence,  it  is  equal  to  the  one-millionth  part  of  the 
weight  of  a  cubic  meter  of  pure  rain-water.  It  is 
called  a  Gram,  and  is  equal  to  15.423  grains  Troy, 
which  is  equal  to  .03527  ounces  Avoirdupois,  very 
nearly.  All  the  other  denominations  are  formed  from 
the  base  as  before.     Hence,  we  have : 


90  MATHEMATICS, 


Ascending  Scale. 

< 

Descending  Scale. 

r—  ■ 

r— i 

uintal 

yriagram 

ilogram 

1 

S-. 

o 
o 

s 

3 

g 

03 
*o 

CD 

g 

o3 

53 

0 

I 

<y  a    m 

►t1 

ft 

6 

Q 

o 

| 

NATURE     OF     THE    METRIC     SYSTEM. 

§  131.  The  Metric  System  is  based  on  the  Meter. 
From  the  Meter,  three  other  units  are  derived  ;  and 
the  four  constitute  the  four  primary  bases  of  this  sys- 
tem of  weights  and  measures.     They  are 

Meter  =  39.37  inches  ;  unit  of  length. 

Are      =  a  square  on  10  meters ;  unit  of  surface. 

Leter  =;  a  cube  whose  edge  is  a  decimeter ;  unit 

of  volume. 
Gram  =  weight  of  a  cube  of  rain-water,  each  edge 

of  which  is  a  centimeter ;  unit  of  weight. 

From  these  four  units  all  the  others  are  derived, 
according  to  the  decimal  scale,  as  already  explained. 
All,  therefore,  which  is  necessary  to  a  full  and  com- 
plete apprehension  of  the  Metric  System,  is  to  com- 
prehend the  meaning  of  the  four  names,  Meter,  Are, 
Leter,  and  Gram — of  the  Latin  prefixes,  Deci,  Centi, 
Milli,  of  the  descending  scale — and  of  the  Greek  pre- 
fixes, Deca,  Hecto,  Kilo,  Myria,  of  the  ascending  scale; 


¥  B  ACIONAL     UXITS.  91 

and  then  remember,  that  every  change  from  one  unit 
to  another,  is  according  to  the  scale  of  tens.  This 
system,  therefore,  requires  only  the  knowledge  of 
eleven  words,  and  of  only  one,  and  that,  the  simplest, 
law  of  number,  viz.,  the  change  from  one  unit  to 
another  according  to  the  scale  of  tens.  It  would  seem 
impossible,  that  so  simple  a  system  of  computation  and 
record,  forming  a  common  language  for  the  whole 
family  of  man,  and  reaching  every  operation  of  trade 
and  commerce,  should  not,  at  an  early  day,  become 
universal. 


FKACTIONAL    UNITS. 


FRACTIONAL     UNITS. SCALE     OF    TENS. 

§  132.  If  the  unit  1  be  divided  into  ten  equal 
parts,  each  part  is  called  one  tenth.  If  one  of  these 
tenths  be  divided  into  ten  equal  parts,  each  part  is 
called  one  hundredth.  If  one  of  the  hundredths  be 
divided  into  ten  equal  parts,  each  part  is  called  one 
thousandth  j  and  corresponding  names  are  given  to 
similar  parts,  how  far  soever  the  divisions  may  be 
carried. 

Now,  although  the  tenths  which  arise  from  divid- 
ing the  unit  1,  are  but  equal  parts  of  1,  they  are, 


92  MATHEMATICS 


nevertheless,  whole  tenths,  and  in  this  light  may  be 
regarded  as  units. 

To  avoid  confusion,  in  the  use  of  terms,  we  shall 
call  every  equal  part  of  1,  a  fractional  unit  Hence, 
tenths,  hundredths,  thousandths,  tenths  of  thousandths, 
&c,  are  fractional  units,  each  having  a  fixed  relation 
to  the  unit  1,  from  which  it  was  derived. 

§  133.  Adopting  a  similar  language  to  that  used  in 
integral  numbers,  we  call  the  tenths,  fractional  units 
of  the  first  order  /  the  hundredths,  fractional  units  of 
the  second  order ;  the  thousandths,  fractional  units 
of  the  third  order  /  and  so  on  for  the  subsequent 
divisions. 

Is  there  any  arithmetical  language  by  which  these 
fractional  nnits  may  be  expressed  ?  The  decimal 
point,  which  is  merely  a  dot,  or  period,  indicates  the 
division  of  the  unit  1,  according  to  the  scale  of  tens. 
By  the  arithmetical  language,  the  unit  of  the  place 
next  the  point,  on  the  right,  is  1  tenth ;  that  of  the 
second  place,  1  hundredth;  that  of  the  third,  1  thou- 
sandth; that  of  the  fourth,  1  ten -thousandth;  and  so 
on  for  places  still  to  the  right. 

The  scale  for  decimals,  therefore,  is 

.000  0  0  00  0  0,  &c; 

in  which  the  unit  of  each  place  is  known  as  soon  as  we 
have  learned  the  signification  of  the  language. 


FRACTIONAL     UNITS.  93 

If,  therefore,  we  wish  to  express  any  of  the  parts 
into  which  the  unit  1  may  be  divided,  according  to 
the  scale  of  tens,  we  have  simply  to  select  from  the 
alphabet,  the  figure  that  will  express  the  number  of 
parts,  and  then  write  it  in  the  place  corresponding  to 
the  order  of  the  unit.  Thus,  to  express  four  tenths, 
three  thousandths,  eight  ten-thousandths,  and  six  mil- 
lion ths,  we  write 

.403806; 
and  similarly,  for  any  decimal  which  can  be  named. 

§  134.  It  should  be  observed  that  while  the  units 
of  place  decrease,  according  to  the  scale  of  tens,  from 
left  to  right,  they  increase  according  to  the  same 
scale,  from  right  to  left.  This  is  the  same  law  of 
increase  as  that  which  connects  the  units  of  place  in 
abstract  numbers.  Hence,  abstract  numbers  and  de- 
cimals being  formed  according  to  the  same  law,  may 
be  written  by  the  side  of  each  other,  and  treated  as 
a  single  number,  by  merely  preserving  the  separat- 
ing or  decimal  point.  Thus,  8974  and  .67046  may 
be  written 

8974.67046; 

since  ten  units,   in   the    place   of    tenths,   make   the 
unit  one,  in  the  place  next  to  the  left. 


94  MATHEMATICS. 


FRACTIONAL      UNITS     IN     GENERAL. 

§  135.  If  the  unit  1  be  divided  into  two  equal  parts, 
each  part  is  called  a  half.  If  it  be  divided  into  three 
equal  parts,  each  part  is  called  a  third :  if  it  be  divided 
into  four  equal  parts,  each  part  is  called  a  fourth :  if 
into  five  equal  parts,  each  part  is  called  a  fifth;  and 
if  into  any  other  number  of  equal  parts,  a  name  is 
given  corresponding  to  the  number  of  parts. 

Now,  although  these  halves,  thirds,  fourths,  fifths, 
&c,  are  each  but  parts  of  the  unit  1,  they  are, 
nevertheless,  in  themselves,  whole  things.  That  is, 
a  half,  is  a  whole  half;  a  third,  a  whole  third;  a 
fourth,  a  whole  fourth;  and  the  same  for  any  other 
equal  part  of  1.  In  this  sense,  therefore,  they  are 
units,  and  we  call  them  fractional  units.  Each  is  an 
exact  part  of  the  unit  1,  and  has  a  fixed  relation  to  it. 

§  136.  Is  there  any  arithmetical  language  by  which 
these  fractional  units  can  be  expressed? 

The  bar  written  at  the  right,  is  the  sign 
which  denotes  the  division  of  the  unit  1 
into    any   number   of  equal   parts. 

If  we  wish   to   express  the  number   of  equal   parts 
into  which  it  is  divided,  as  9,  for  example, 
we  simply  write  the  9   under  the  bar,  and  Q 

then  the    phrase    means,    that    some    thing 


FRACTIONAL    UNITS.  95 

regarded  as   a   whole,   has  been   divided  into   9  equal 
parts. 

If,    now,    we    wish    to    express    any    number     of 
these    fractional   units,   as    7,   for    example, 
wTe  place   the   7  above  the   line,    and   read, 
seven   ninths. 


§  137.  It  has  been  observed,  that  two  things  are 
necessary  to  the  clear  apprehension  of  an  integral 
number. 

1st.  A  distinct  apprehension  of  the  unit  which 
forms   the  base   of  the  number;    and, 

2d.  A  distinct  apprehension  of  the  number  of  times 
which  that  unit  is  taken. 

Three  things  are  necessary  to  the  distinct  appre- 
hension of  the  value  of  any  fraction,  either  decimal 
or  vulgar. 

1st.  We  must  know  the  unit,  or  wdiole  thing, 
from  which  the  fraction  was  derived ; 

2d.  We  must  know  into  how  many  equal  parts 
that  unit  is  divided  ;   and, 

3d.  We  must  know  how  many  such  parts  are  taken 
in   the  expression. 

The  unit  from  which  the  fraction  is  derived,  is 
called  the  unit  of  the  fraction  /and  one  of  the 
equal  parts  is  called,  the  fractional  unit. 

For  example,  to  apprehend  the  value  of  the  fraction 
f  of  a  pound  avoirdupois,  or  -| lb. ;  we  must  know, 


96  MATHEMATICS. 

1st.   "What  is  meant  by   a  pound  ; 

2d.    That   it   has    been    divided    into    seven    equal 
parts;   and, 
,    3d.   That   three  of  those  parts   are   taken. 

In  the  above  fraction,  1  pound  is  the  unit  of 
the  fraction ;  one-seventh  of  a  pound,  is  the  frac- 
tional unit;  and  3,  denotes  that  three  fractional  units 
are  taken. 

If  the  unit  of  a  fraction  is  not  named,  it  is 
taken  to  be  the  abstract  unit  1. 

ADVANTAGES     OF     FRACTIONAL      UNITS. 

§  138.  By  considering  every  equal  part  of  a  unit 
as  an  entire  thing,  having  a  certain  relation  to  the 
unit  1,  the  mind  is  led  to  analyze  a  fraction,  and 
.thus  to  apprehend  its  precise  signification. 

Under  this  searching  analysis,  the  mind  at  once 
seizes  on  the  unit  of  the  fraction  as  the  principal 
base.  It  then  looks  at  the  value  of  each  part.  It 
then   inquires  how   many   such   parts   are   taken. 

It  having  been  shown  that  equal  integral  units 
can  alone  be  added,  it  is  readily  seen  that  the  same 
principle  is  equally  applicable  to  fractional  units ; 
and  then  the  inquiry  is  made :  What  is  necessary 
in  order  to  make  such   units  equal  ? 

It  is  seen  at  once,  that  two  things  are  necessary : 

1st.  That  they  be  parts  of  the  same  unit  *   and, 


FRACTIONAL    UNITS.  97 

2d.  That  they  be  like  jparts ;  in  other  words,  they 
must  be  of  the  same  denomination,  and  have  a  com- 
mon denominator. 

In  regard  to  Decimal  Fractions,  all  that  is  neces- 
sary, is  to  observe  that  units  of  the  same  value  are 
added  to  each  other,  and  when 'the  figures  expressing 
them  are  written  down,  they  should  always  be  placed 
in  the  same  column. 

§  139.  The  great  difficulty  in  the  management  of 
fractions,  consists  in  comparing  them  with  each  other, 
instead  of  constantly  comparing  them  with  the  unit 
from  which  they  are  derived.  By  considering  them  as 
entire  things,  having  a  fixed  relation  to  the  unit  which 
is  their  base,  they  can  be  compared  as  readily  as 
integral  numbers ;  for,  the  mind  is  never  at  a  loss 
when  it  apprehends  the  unit,  the  parts  into  which  it 
is  divided,  and  the  number  of  parts  which  are  taken. 
The  only  reasons  why  we  apprehend  and  handle  in- 
tegral numbers  more  readily  than  fractions,  are, 

1st.  Because  the  unit  forming  the  base  is  always 
kept  in  view;   and, 

2d.  Because,  in  integral  numbers,  we  have  been 
taught  to  trace  constantly  the  connection  between  the 
unit  and  the  numbers  which  come  from  it;  while  in 
the  methods  of  treating  fractions,  these  important 
considerations  have  been  neglected. 


98  MATHEMATICS, 


PROPORTION    AND  RATIO. 

§  140.  Proportion  expresses  the  relation  which  one 
number  hears  to  another,  with  respect  to  its  "being 
greater  or  less. 

Two  numbers  may  he  compared,  the  one  with  the 
other,  in  two  ways : 

1st.  With  respect  to  their  difference,  called  Arith- 
metical Proportion;   and,  # 

2d.  With  respect  to  their  quotient,  called  Geomet- 
rical Proportion. 

Tims,  if  we  compare  the  numbers  1  and  8,  by  their 
difference,  we  find  that  the  second  exceeds  the  first 
by  7 :  hence,  their  difference  7,  is  the  measure  of 
their  arithmetical  relation,  and  is  called  their  arith- 
metical proportion. 

If  we  compare  the  same  numbers  by  their  quotient, 
we  find  that  the  second  contains  the  first  8  times - 
hence,  8  is  the  measure  of  their  geometrical  relation, 
and  is  called  their  ratio. 

§  141.  The  two  numbers  which  are  thus  compared, 
are  called  terms.  The  first  is  called  the  antecedent , 
and  the  second  the  consequent. 

In  comparing  numbers  with  respect  to  their  differ- 
ence, the  question  is,  how  much  is  one  greater  or  less 
than  the  other  ?  Their  difference  affords  the  true  an- 
swer, and  is  the  measure  of  their  proportion. 


PROPORTION    AND    RATIO.  99 

In  comparing  numbers  with  respect  to  their  quo- 
tient, the  question  is,  how  many  times  is  one  greater 
or  less  than  the  other?  Their  quotient  or  ratio,  is  the 
true  answer,  and  is  the  measure  of  their  proportion. 
Ten,  for  example,  is  nine  greater  than  one,  if  we  com- 
pare the  numbers  one  and  ten  by  their  difference. 
But  if  we  compare  them  by  their  quotient,  ten  is  said 
to  be  ten  times  as  great  as  one — the  language  "  ten 
times"  having  reference  to  the  quotient,  which  is  al- 
ways taken  as  the  measure  of  the  relative  value  of 
the  two  numbers  so  compared.  Thus,  when  we  say, 
that,  the  units  of  our  common  system  of  numbers 
increase  in  a  tenfold  ratio,  we  mean  that  they  so  in- 
crease that  each  succeeding  unit  shall  contain  the 
preceding,  one  ten  times.  This  is  a  convenient  lan- 
guage to  express  a  particular  relation  of  two  num- 
bers, and  is  perfectly  correct,  when  used  in  conformity 
to  an  accurate  definition. 

§  142.  The  first  term  is  called,  the  standard ;  and 
is  always  the  unit  of  measure  which  measures  the 
other. 

§  143.  It  may  be  proper  here  to  observe,  that  while 
the  term  "  geometrical  proportion"  is  used  to  express 
the  relation  of  two  numbers,  compared  by  their  ratio, 
the  term  "A  geometrical  proportion,"  is  applied  to 
four  numbers,  in  which  the  ratio  of  the  first  to  the 


100  MATHEMATICS. 

second  is  the  same  as  that  of  the  third  to  the  fourth. 

Thus, 

2  :  4  : :  6  :  12, 

is  a  geometrical  proportion,  of  which  the  ratio  is  2. 

§  144.  One  important  remark  on  the  subject  of 
proportion  is  yet  to  be  made.     It  is  this : 

Any   two    numbers    which    are    compared  together, 
either  by  their  difference  or  quotient,  must  be  of  the' 
same  hind:   that  is,  they  must  either  have  the  same 
unity  as  a  base,  or  be  susceptible  of  reduction  to  the 
same  unit. 

For  example,  we  can  compare  2  pounds  with  6 
pounds  :  their  difference  is  4  pounds,  and  their  ratio 
is  the  abstract  number  3.  We  can  also  compare  2 
feet  with  8  yards :  for,  although  the  unit  1  foot  is 
different  from  the  unit  1  yard,  still  8  yards  are  equal 
to  24  feet.  Hence,  the  difference  of  the  numbers  is 
22  feet,  and  their  ratio  the  abstract  number  12. 

On  the  other  hand,  we  cannot  compare  2  dollars 
with  2  yards  of  cloth,  for  they  are  quantities  of 
different  kinds,  not  being  susceptible  of  reduction  to 
a  common  unit. 

Simple  or  abstract  numbers  may  always  be  com- 
pared, since  they  have  a  common  unit  1. 


SECTION    IV, 

GEOMETRY. 

\ 


§  145.  Geometry  treats  of  space,  and  compares 
portions  of  space  with  each  other,  for  the  purpose 
of  pointing  out  their  properties  and  mutual  rela- 
tions. The  science  consists  in  the  development  of 
all  the  laws  relating  to  space,  and  is  made  up  of 
the  processes  and  rules,  by  means  of  which  por- 
tions of  space  can  be  best  compared  with  each 
other.  The  truths  of  Geometry  are  a  series  of  de- 
pendent propositions,  and  may  be  divided  into  three 
classes : 

1st.  Truths  implied  in  the  definitions,  viz.  that 
things  do  exist,  or  may  exist,  corresponding  to  the 
words  defined.  For  example,  when  we  say,  "  A 
quadrilateral  is  a  rectilinear  figure  having  four  sides," 
we  imply  the  existence  of  such  a  figure. 

2d.  Self-evident,  or  intuitive  truths,  embodied  in 
the   axioms;    and, 

3d.  Truths  inferred  or  proved,  from  the  definitions 
and  axioms,   called  Demonstrative   Truths.     We  say 


102  MATHEMATICS. 

that  a  truth  or  proposition  is  proved  or  demon- 
strated, when,  by  a  course  of  reasoning,  it  is  shown 
to  be  included  under  some  other  truth  or  propo- 
sition, previously  known,  and  from  wliich  it  is  said 
to  be  derived ;  hence, 

A  Demonstration  is  a  series  of  logical  arguments, 
brought  to  a  conclusion,  in  which  the  major  pre- 
mises are  definitions,  axioms,  or  propositions  already 
established. 

§  146.  JBefore  we  can  understand  the  proofs  or 
demonstrations  of  Geometry,  wre  must  understand 
what  that  is  to  which  demonstration  is  applicable: 
hence,  the  first  thing  necessary  is  to  form  a  clear 
conception  of  space,  the  subject  of  all  geometrical 
reasoning. 

The  next  step  is  to  give  names  to  particular  forms 
or  limited  portions  of  space,  and  to  define  these 
names  accurately.  The  definitions  of  these  names 
are  the  definitions  of  Geometry,  and  the  portions 
of  space  corresponding  to  them  are  called  Figures, 
or  Geometrical  Magnitudes;  of  which  there  are  four 
general   classes : 

1st.   Lines  ; 

2d.   Surfaces ; 

3d.   Volumes ;   and. 

4th.  Angles. 


GEO  MET  BY.  103 


§  147.  Lines  embrace  only  one  dimension  of  space, 
viz.  length,  without  breadth  or  thickness.  The  ex- 
tremities, or  limits  of  a  line,  are  called  points. 

There  are  two  general  classes  of  lines — straight 
lines  and  curved  lines.  A  straight  line  is  one  which 
lies  in  the  same  direction  between  any  two  of  its 
points  ;  and  a  curved  line  is  one  which  constantly 
changes  its  direction  at  every  point.  There  is  but 
one  kind  of  straight  line,  and  that  is  fully  char- 
acterized by  the  definition.  From  the  definition  we 
may  infer  the  following  axiom :  "  A  straight  line 
is  the  shortest  distance  between  two  points."  There 
are  many  kinds  of  curves,  of  which  the  circum- 
ference of  the  circle  is  the  simplest  and  the  most 
easily   described. 

§  148.  Surfaces  embrace  two  dimensions  of  space, 
viz.  length  and  breadth,  but  not  thickness.  Sur- 
faces, like  lines,  are  also  divided  into  two  general 
classes,  viz.  plane  surfaces  and  curved  surfaces. 

A  plane  surface  is  that  with  which  a  straight 
line,  any  how  placed,  and  having  two  points  com- 
mon with  the  surface,  will  coincide  throughout  its 
entire  extent.  Such  a  surface  is  perfectly  even,  and 
is  commonly  designated  by  the  term  "  A  plane." 
A  large  class  of  the  figures  of  Geometry  are  but 
portions  of  a  plane,  and  all  such  are  called  plane 
figures. 


104  MATHEMATICS 


§  149.  A  portion  of  a  plane,  bounded  by  three 
straight  lines,  is  called  a  triangle,  and  this  is  the 
simplest  of  the  plane  figures.  There  are  several  kinds 
of  triangles,  differing  from  each  other,  however,  only 
in  the  relative  values  of  their  sides  and  angles.  For 
example:  when  the  sides  are  all  equal  to  each  other, 
the  triangle  is  called  equilateral ;  when  two  of  the 
sides  are  equal,  it  is  called  isosceles ;  and  scalene, 
when  the  three  sides  are  all  unequal.  If  one  of  the 
angles  is  a  right  angle,  the  triangle  is  called  a  right- 
angled  triangle. 

§  150.  The  next  simplest  class  of  plane  figures  com- 
prises all  those  which  are  bounded  by  four  straight 
lines,  and  are  called  quadrilaterals.  There  are  several 
varieties  of  this  class : 

1st.  The  mere  quadrilateral,  which  has  no  mark, 
except  that  of  having  four  sides*  called  a  trape- 
zium; 

2d.  The  trapezoid,  which  has  two  sides  parallel 
and  two  not  parallel; 

3d.  The  parallelogram,  which  has  it  opposite  sides 
parallel  and  its  angles  oblique  ; 

4th.  The  rectangle,  which  has  all  its  angles  right 
angles  and  its  opposite  sides  parallel;  and, 

5th.  The  square,  which  has  its  four  sides  equal  to 
each  other,  each  to  each,  and  its  four  angles  right 
angles. 


GEOMETRY.  105 


§  151.  Plane  figures,  bounded  by  straight  lines, 
having  a  number  of  sides  greater  .than  four,  take 
names  corresponding  to  the  number  of  their  sides, 
viz.  Pentagons,  Hexagons,  Heptagons,  &c,  and  the 
common  term,  Polygon,  is  applicable  to  every  plane 
figure  bounded  by  straight  lines. 


§  152.  A  portion  of  a  plane  bounded  by  a  curved 
line,  all  the  points  of  which  are  equally  distant  from 
a  certain  point  within  called  the  centre,  is  called 
a  circle,  and  the  bounding  line  is  called  the  circum- 
ference. This  is  the  only  curve  usually  treated  of 
in  Elementary  Geometry. 

§  1 53.  A  curved  surface,  like  a  plane,  embraces 
the  two  dimensions  of  length  and  breadth.  It  is  not 
even,  like  the  plane,  throughout  its  whole  extent,  and 
therefore  a  straight  line  may  have  two  points  in 
common,  and  yet  not  coincide  with  it.  The  surface 
of  the  cone,  of  the  sphere,  and  cylinder,  are  the 
curved  surfaces  treated  of  in  Elementary  Geometry. 

§  154.  A  volume  is  a  portion  of  space,  combining 
the  three  dimensions  of  length,  breadth,  and  thick- 
ness.    Volumes  are  divided  into  three  classes: 

1st.  Those  bounded  by  planes ; 

2d.  Those  bounded  by  plane  and  curved  surfaces; 

and, 

5* 


106  MATHEMATICS. 

3d.  Those  bounded  only  by  curved  surfaces. 

The  first  class  embraces  the  pyramid  and  prism 
with  their  several  varieties ;  the  second  class  embraces 
the  cylinder  and  cone;  and  the  third  class  the  sphere, 
together  with  others  not  generally  treated  of  in 
Elementary  Geometry. 

§  155.  An  angle,  in  Elementary  Geometry,  is  the 
amount  of  inclination  of  two  lines,  or  of  two  planes. 
Angles  are  generally  compared  with  the  right  angle^ 
which  is  their  natural  unit. 

§  156.  We  have  now  named  all  the  geometrical 
magnitudes  treated  of  in  Elementary  Geometry.  They 
are  merely  limited  portions  of  space,  and  do  not, 
necessarily,  involve  the  idea  of  matter.  A  sphere, 
for  example,  fulfils  all  the  conditions  imposed  by  its 
definition,  without  any  reference  to  what  may  be 
within  the  space  enclosed  by  its  surface.  That  space 
may  be  occupied  by  lead,  iron,  or  air,  or  may  be 
a  vacuum,  without  at  all  changing  the  nature  of  the 
sphere,  as  a  geometrical  magnitude. 

It  should  be  observed  that  the  boundary  or  limit 
of  a  geometrical  magnitude,  is  another  geometrical 
magnitude,  having  one  dimension  less.  For  example : 
the  boundary  or  limit  of  a  volume,  which  has  three 
dimensions,  is  always  a  surface  which  has  but  two: 
the    limits    or    boundaries    of   all    surfaces   are  lines, 


GEOMETEY.  107 


straight  or  curved ;   and  the  extremities  or  limits  of 
lines  are  points. 

§  157.  We  have  now  named  and  shown  the  nature 
of  the  tilings  which  are  the  subjects  of  Elementary 
Geometry.  They  are  called,  the  Geometrical  Magni- 
tudes. The  science  of  Geometry  is  a  collection  of 
those  connected  processes  by  which  we  determine  the 
measures,  properties,  and  relations  of  these  magni- 
tudes. 


COMPARISON    OF   FIGURES    WITH    UNITS    OF    MEASURE. 

§  158.  We  have  seen  that  the  term  measure,  im- 
plies a  comparison  of  the  thing  measured  with  some 
known  thing  of  the  same  kind,  regarded  as  a  standard; 
and  that  such  standard  is  called  the  unit  of  measure. 
The  unit  of  measure  for  lines  must,  therefore,  be  a 
line  of  a  known  length ;  a  foot,  a  yard,  a  rod,  a  mile, 
or  any  other  known  unit. 

The  unit  of  measure  for  surfaces,  is  a  square  con- 
structed on  the  linear  unit  as  a  side :  that  is,  a  square 
foot,  a  square  yard,  a  square  rod,  a  square  mile ;  that 
is,  a  square  described  on  any  known  unit  of  length. 

The  unit  of  measure,  for  volumes,  is  a  volume,  and 
therefore  has  three  dimensions.  It  is  a  cube  con- 
structed on  a  linear  unit  as  an  edge,  or  on  the  super- 
ficial unit  as  a  base.     It  is,  therefore,  a  cubic  foot,  a 


108  MATHEMATICS. 

cubic  yard,  a  cubic  rod,  &c.  The  unit  of  measure  for 
angles  is  the  right  angle.  With  this  standard  all 
angles  are  compared.  Hence,  there  are  four  units  of 
measure,  each  differing  in  kind  from  the  other  two, 
viz.  a  known  length  for  the  measurement  of  lines  ;  a 
known  square  for  the  measurement  of  surfaces ;  a 
known  volume  for  the  measurement  of  volumes  ;  and  a 
known  angle  for  the  measurement  of  angles.  The 
measure  or  contents  of  any  magnitude,  belonging  to 
either  class,  is  ascertained  by  finding  how  many 
times  that  magnitude  contains  its  unit  of  measure. 

§  159.  "We  have  dwelt  with  much  detail  on  the 
unit  of  measure,  because  it  furnishes  the  only  base  of 
estimating  quantity.  The  conception  of  number  and 
space  merely  opens  to  the  intellectual  vision  an  un- 
measured field  of  investigation  and  thought,  as  the 
ascent  to  the  summit  of  a  mountain  presents  to  the 
eye  a  wide  and  un surveyed  horizon.  To  ascertain 
the  height  of  the  point  of  view,  the  diameter  of  the 
surrounding  circular  area  and  the  distance  to  any 
point  which  may  be  seen,  some  standard  or  unit 
must  be  known,  and  its  value  distinctly  apprehended. 
So,  also,  number  and  space,  which  at  first  .fill  the  mind 
with  vague  and  indefinite  conceptions,  are  to  be  finally 
measured  by  units  of  ascertained  value. 

§  160.   It  is  found,  on   careful  analysis,  that  every 


GEOMETRY.  109 


number  may  be  referred  to  the  unit  one,  as  a  stan- 
dard, and  when  the  signification  of  the  term  one  is 
clearly  apprehended,  that  any  number,  whether  large 
or  small,  whether  integral  or  fractional,  may  be  de- 
duced from  the  standard  by  an  easy  and  known 
process. 

In  space,  also,  which  is  indefinite  in  extent,  and 
exactly  similar  in  all  its  parts,  the  faculties  of  the 
mind  have  established  ideal  boundaries.  These  boun- 
daries give  rise  to  the  geometrical  magnitudes,  each 
of  which  has  its  own  unit  of  measure ;  and  by  these 
simple  contrivances,  we  measure  space,  even  to  the 
stars,  as  with  a  yardstick. 

§  161.  We  have,  thus  far,  not  alluded  to  the  diffi- 
culty of  determining  the  exact  length  of  that  which 
we  regard  as  a  standard.  We  are  presented  with 
a  given  length,  and  told  that  it  is  called  a  foot  or 
a  yard  ;  and  this  being  usually  done  at  a  period  of 
life  when  the  mind  is  satisfied  with  mere  facts,  we 
adopt  the  conception  of  a  distance  corresponding  to 
a  name,  and  then  by  multiplying  and  dividing  that 
distance  we  are  enabled  to  apprehend  other  distances. 
But  this  by  no  means  answers  the  inquiry,  What  is 
the  standard  for  measurement? 

Under  the  supposition  that  the  laws  of  physi- 
cal nature  operate  uniformly,  the  unit  of  measure 
in  England   and  the    United    States  has  been   fixed 


110  MATHEMATIC 


by  ascertaining  the  length  of  a  pendulum  which 
will  vibrate  seconds,  and  to  this  length  the  Im- 
perial yard,  which  we  have  also  adopted  as  a 
standard,  is  referred.  Hence  the  unit  of  measure 
is  referred  to  a  natural  standard,  viz.  to  the  distance 
between  the  axis  of  suspension  and  the  centre  of 
oscillation  of  a  pendulum  which  shall  vibrate  sec- 
onds in  vacuo,  in  London,  at  the  level  of  the  sea. 
This  distance  is  declared  to  be  39.1393  imperial 
inches/  that  is,  3  imperial  feet  and  3.1393  inches. 
Tims,  the  determination  of  the  unit  of  length  de- 
mands the  application  of  the  most  abstruse  science, 
combined  w7ith  accurate  observation  and  delicate 
experiment. 

Could  this  distance,  or  unit,  have  been  exactly 
ascertained  before  the  measures  of  the  world  were 
fixed,  and  in  general  use,  it  would  have  afforded 
a  standard  at  once  certain  and  convenient,  and  all 
distances  would  then  have  been  expressed  in  num- 
bers arising  from  its  multiplication  or  exact  divi- 
sion. But  as  the  measures  of  the  world  (and  con- 
sequently their  units)  were  fixed  antecedently  to 
the  determination  of  this  distance,  it  was  expressed 
in  measures  already  known ;  and  hence,  instead  of 
being  represented  by  1,  which  had  already  been 
appropriated  to  the  foot,  it  was  expressed  in  terms 
of  the  foot,  viz.  39.1393  inches,  and  this  is  now  the 
standard  to  which  all  units  of  measure  are  referred. 


GEOMETRY.  Ill 


§  162.  rJlie  French  Government,  in  order  to  obtain 
an  invariable  unit,  measured  a  degree  of  the  arc  of 
a  meridian  on  the  earth's  surface ;  and  from  this 
computed  the  length  of  the  meridional  arc  from  the 
equator  to  the  pole.  This  length  they  divided  into  ten 
million  equal  parts,  and  then  took  one  of  these  parts 
for  the  unit  of  length,  and  called  it  a  Meter.  The 
length  of  this  meter  is  equal  to  1  yard  3  inches 
and  37  hundredths  of  an  inch,  very  nearly.  Thus 
they  obtained  the  length  of  the  unit  which  is  the 
base  of  the  Metric  System  of  "Weights  and  Measures. 
The  next  step  was  to.  fix  the  law  by  which  the 
other  units  of  the  system  should  be  obtained  from 
the  base.  As  the  scale  of  tens  is  the  simplest  law 
by  which  we  can  pass  from  one  unit  to  another, 
that  scale  was  adopted,  and  the  larger  units  are 
formed  by  multiplying  the  base  continually  by  10, 
and  the  smaller,  by   dividing  it  continually  by  10. 

The  Metric  System  of  Weights  and  Measures,  by 
furnishing  a  com morr  language  to  all  nations,  applica- 
ble to  the  measure,  cost,  and  value  of  every  produc- 
tion of  labor,  of  art,  and  of  commerce,  will,  if  uni- 
versally adopted,  go  far  towards  bringing  about  en- 
tire uniformity;  and  will  produce  similar  changes  in 
the  department  of  industry,  to  those  which  the  tele- 
graph has  produced  in  the  department  of  thought. 

§  163.  The  unit  of  measure,  whether  it  be  the  im- 


112  MATHEMATICS. 

perial  yard  or  the  French  meter,  is  not  only  impor- 
tant as  affording  a  basis  for  all  measurement,  but  is 
also  the  element  from  which  we  deduce  the  unit 
of  weight.  The  weight  of  27.7015  cubic  inches  of 
distilled  water  is  taken  as  the  'English  standard  of 
weight,  weighing  exactly  one  pound  avoirdupois,  and 
this  quantity  of  water  is  determined  from  the  unit  of 
length  ;  that  is,  the  determination  of  it  reaches 
back  to  the  length  of  a  pendulum  which  will  vi- 
brate seconds  in  the  latitude  of  London. 

§  164.  Two  geometrical  figures  are  said  to  be  equal, 
when  they  contain  the  same  unit  of  measure  an 
equal  number  of  times.  Two  figures  are  said  to  be 
equal  in  all  their  parts,  when  they  can  be  so  applied 
to  each  other  as  to  coincide  throughout  their  whole 
extent.  Hence,  equal  refers  to  measure,  and  equality 
in  all  their  parts,  to  coincidence.  Indeed,  coincidence 
is  the  only  test  of  perfect  geometrical  equality. 


PROPERTIES     OF     FIGURES. 

§  165.  A  property  of  a  figure  is  a  mark  common 
to  all  figures  of  the  same  class.  For  example ;  if 
the  class  be  "  Quadrilateral,"  there  are  two  very 
obvious  properties,  common  to  all  quadrilaterals,  be- 
sides the  one  which  characterizes  the  figure,  and  by 
which  its  name  is  defined,  viz.  that  it  has  four  angles, 


GEOMETRY.  113 


and  that  it  may  de  divided  into  two  triangles.  If  the 
class  be  "Parallelogram,"  there  are  several  properties 
common  to  all  parallelograms,  and  which  are  subjects 
of  proof;  such  as,  that  the  opposite  sides  and  angles 
are* equal;  the  diagonals  divide  each  other  into  equal 
parts,  &g.  If  the  class  be  "  Triangle,"  there  are  many 
properties  common  to  all  triangles,  besides  the  char- 
acteristic that  each  has  three  sides.  If  the  class  be 
a  particular  kind  of  triangle,  such  as  the  equilateral, 
isosceles,  or  right-angled  triangle,  then  each  class  has 
particular  properties,  common  to  every  individual  of 
the  class,  but  not  common  to  the  other  classes. 

It  is  important,  however,  to  remark,  that  every 
property  which  belongs  to  "triangle,"  regarded  as  a 
genus,  will  appertain  to  every  species  or  class  of  tri- 
angle ;  and  universally,  every  property  which  belongs 
to  a  genus  will  belong  to  every  species  under  it ;  and 
every  property  which  belongs  to  a  species  will  be- 
long to  every  class  or  subspecies  under  it;  and  every 
property  which  belongs  to  one  of  a  subspecies  or 
class  will  be  common  to  every  individual  of  the  class. 
For  example:  "the  square  on  the  hypothenuse  of  a 
right-angled  triangle  is  equal  to  the  sum  of  the 
squares  described  on  the  other  two  sides,"  is  a  pro- 
position equally  true  of  every  right-angled  triangle: 
and  "every  straight  line  perpendicular  to  a  chord, 
at  the  middle  point,  will  pass  through  the  centre," 
is  equally  true  of  all  chords  and  of  all  circles. 


114  MATHEMATICS, 


MARKS    OF    WHAT    MAT    BE    PROVED. 

§  166.  The  characteristic  properties  of  every  ge- 
ometrical figure  (that  is,  those  properties  without 
which  the  figures  could  not  exist),  are  given  in  the 
definitions.  How  are  we  to  arrive  at  all  the  other 
properties  of  these  figures?  The  propositions  implied 
in  the  definitions,  viz.  that  things  corresponding  to 
the  words  defined  do  or  may  exist  with  the  proper- 
ties named  ;  and  the  self-evident  propositions  or  ax- 
ioms, contain  the  only  marks  of  what  can  be  proved; 
and  by  a  skilful  combination  of  these  marks  we  are 
able  to  discover  and  prove  all  that  is  discovered  and 
proved  in  Geometry. 

Definitions  and  axioms,  and  propositions  deduced 
from  them,  are  major  premises  in  each  new  demon- 
stration; and  the  science  is  made  up  of  the  processes 
employed  for  bringing  unforeseen  cases  under  these 
known  truths;  or,  in  syllogistic  language,  for  prov- 
ing the  minors  necessary  to  complete  the  syllogisms. 
The  marks  being  so  few,  and  the  inductions  which 
furnish  them  so  obvious  and  familiar,  there  would 
seem  to  be  very  little  difficulty  in  the  deductive  pro- 
cesses which  follow.  The  connecting  together  of 
several  of  these  marks  are  Deductions,  or  Trains  of 
Reasoning ;  and  hence,  Geometry  is  a  Deductive 
Science. 


GEOMETRY 


115 


DEMONSTRATION. 


§  167.  As  a  first  example,  let  us  take  the  first 
proposition  in  Legendre's  Geometry: 

"  If  a  straight  line  onset  another  straight  line,  the 
sum  of  the  two  adjacent  angles  will  oe  equal  to  two 
right  angles" 

Let  the  straight  line  DO  meet 
the  straight  line  AB  at  the  point 
C,  then  will  the  angle  ACD  plus 

the  angle  DCB  be  equal  to  two 

right  angles. 

To  prove  this  proposition,  we  need  the  definition 
of  a  right  angle,  viz.  : 

When  a  straight  line  AB  meets 
another  straight  line  CD,  so  as 
to  make  the  adjacent  angles  BAG 

and  BAD  equal  to  each  other,      

each  of  those   angles  is   called  a 
eight  angle,  and  the  line  AB  is  said  to  he  perpen- 
dicular to  CD. 

We  shall  also  need  the  1st,  2d,  and  9th  axioms, 
for  inferring  equality,  viz. : 

1.  Things  which  are  equal  to  the  same  thing  are 
equal  to  each  other. 

2.  If  equals  be  added  to  equals,  the  sums  will  be 
equal. 


D 


U6 


MATHEMATICS. 


9.  A  whole  is  equal  to  the  sum  of  all  its  parts. 

]STow,  before  these  formulas  or  tests  can  be  applied, 
it  is  necessary  to  suppose  a  straight 
line  CE  to  be  drawn  perpendicular 
to    AB    at  the   point    C :   then  by 
the  definition  of  a  right  angle,  the 

angle   ACE   will   be  equal  to  the    

angle  ECB.  A  °  B 

By  axiom  9th,  we  have, 

A  CD  equal  to  ACE  plus  ECD :  to  each  of  these 
equals  add  DCB;  and  by  the  2d  axiom  we  shall 
have, 


!  ACD  plus  DCB  equal  to  ACE  plus  ECD  plus 
DCB ;  but  by  axiom  9th, 

ECD  plus  DCB  equals  ECB;  therefore  by  axiom 
1st, 

ACD  plus  DCB  equals  ACE  plus  ECB. 

But  by  the  definition  of  a  right  angle, 

ACE  plus  ECB  equals  two  right  angles:  therefore, 
by  the  1st  axiom, 

ACD  plus  DCB  equals  two  right  angles. 

It  will  be  seen  that  the  conclusiveness  of  the  proof 
results, 

1st.  From  the  definition,  that  ACE  and  ECB  are 
equal  to  each  other,  and  each  is  called  a  right  angle : 
consequently,  their  sum  is  equal  to  two  right  angles; 
and, 

2d.      In    showing,    by   means    of  the  axioms,  that 


GEOMETRY.  117 


ACD  plus  DCB  equals  ACE  plus  ECB ;  and  then 
inferring  from  axiom  1st,  that  ACD  plus  DCB 
equals  two  right  angles. 

§  168.  The  difficulty  in  the  geometrical  reasoning 
consists  mainly  in  showing  that  the  proposition  to  be 
proved  contains  the  marks  which  prove  it.  To  ac- 
complish this,  it  is  frequently  necessary  to  draw 
many  auxiliary  lines,  forming  new  figures  and  angles, 
which  can  be  shown  to  possess  marks  of  these  marks, 
and  which  thus  become  connecting  links  between 
the  known  and  the  unknown  truths.  Indeed,  most 
of  the  skill  and  ingenuity  exhibited  in  the  geomet- 
rical processes  are  employed  in  the  use  of  these  aux- 
iliary means.  The  example  above  affords  an  illustra- 
tion. 

We  were  unable  to  show  that  the  sum  of  the  two 
angles  possessed  the  mark  of  being  equal  to  two  right 
angles,  until  we  had  drawn  a  perpendicular,  or  sup- 
posed one  drawn,  at  the  point  where  the  given  lines 
intersect.  That  being  done,  the  two  right  angles 
ACE  and  ECB  were  formed  which  enabled  us  to 
compare  the  sum  of  the  angle  ACD  and  DCB  with 
two  right  angles,  and  thus  we  jproved  the  proposi- 
on. 

§  169.  As  a  second  example,  let  us  take  the  fol- 
lowing proposition : 


118 


MATHEMATICS 


If  two  straight  lines  meet  each  other ',  the  opposite 
or  vertical  angles  will  be  equal. 

Let    the    straight    line    AB  A 
meet   the    straight   line   ED    at 
the  point  C:  then  will  the  angle 
ACD  be  equal  to  the  opposite 
angle     ECB;     and    the     angle  E 
ACE  equal  to  the  angle  DCB. 

To  prove  this  proposition,  we  need  the  last  propo- 
sition, and  also  the  1st  and  5th  axioms  of  Legendre, 
viz. : 

"  If  a  straight  line  meet  another  straight  line,  the 
sum  of  the  two  adjacent  angles  will  be  equal  to  two 
right  angles." 

"  Things  which  are  equal  to  the  same  thing  are 
equal  to  each  other." 

"  If  equals  be  taken  from  equals,  the  remainders 
will  be  equal." 

Now,  since  the  straight  line  AC  meets  the  straight 
line  ED  at  the  point  C,  we  have, 

ACD  plus  ACE  equal  to   two  right  angles. 

And  since  the  straight  line  DC  meets  the  straight 
line  AB,  we  have, 

ACD  plus  DCB  equal  to  two  right  angles  :  hence, 
by  the  first  axiom, 

ACD  plus  ACE  equals  ACD  plus  DCB:  taking 
from  each  the  common  angle  ACD,  we  know  from 
the  fifth  axiom  that   the   remainders  will   be  equal  ; 


GEOMETRY.  119 


that  is,  the  angle  ACE  equal  to  the  opposite  or  ver- 
tical angle  DCB. 

§  170.  The  two  demonstrations  given  above  com- 
bine all  the  processes  of  proof  employed  in  every 
demonstration  of  the  same  class.  When  any  new 
truth  is  to  be  proved,  the  known  tests  of  truth  are 
gradually  extended  to  auxiliary  quantities  having  a 
more  intimate  connection  with  such  new  truth  than 
existed  between  it  and  the  known  tests,  until  finally, 
the  known  tests,  through  a  series  of  links,  become 
applicable  to  the  final  truth  to  be  established:  the  in- 
termediate processes,  as  it  were,  bridging  over  the 
space  between  the  known  tests  and  the  final  truth 
to  be  proved. 

§  171.  There  are  two  classes  of  demonstrations, 
quite  different  from  each  other  in  some  respects,  al- 
though the  same  processes  of  argumentation  are  em- 
ployed in  both,  and  although  the  conclusions  in  both 
are  subjected  to  the  same  logical  tests.  They  are 
called  Direct  or  Positive  Demonstration,  and  Nega- 
tive Demonstration,  or  the  Reductio  ad  Absur- 
dum. 

§  172.  The  main  differences  in  the  two  methods 
are  these :  The  method  of  direct  demonstration  rests 
its    arguments    on    known   and    admitted   truths,  and 


120  MATHEMATICS. 

shows  by  logical  processes  that  the  proposition  can 
be  brought  under  some  previous  definition,  axiom,  or 
proposition :  while  the  negative  demonstration  rests 
its  arguments  on  an  hypothesis,  combines  this  with 
"known  propositions,  and  deduces  a  conclusion  by  pro- 
cesses strictly  logical.  Now  if  the  conclusion  so  de- 
duced agrees  with  any  known  truth,  we  infer  that 
the  hypothesis  (which  was  the  only  link  in  the  chain 
not  previously  known)  was  true ;  but  if  the  conclu- 
sion be  excluded  from  the  truths  previously  estab- 
lished ;  that  is,  if  it  be  opposed  to  any  one  of  them, 
then  it  follows  that  the  hypothesis,  being  contradic- 
tory to  such  truth,  must  be  false.  In  the  negative 
demonstration,  therefore,  the  conclusion  is  compared 
with  the  truths  known  antecedently  to  the  proposi- 
tion in  question  :  if  it  agrees  with  any  one  of  them, 
the  hypothesis  is  correct ;  if  it  disagrees  with  any 
one  of  them,  the  hypothesis  is  false. 

§  173.  We  will  give  as  an  illustration  of  this 
method,  the  demonstration  employed  in  Proposition 
XVII.  of  the  First  Book  of  Legendre: 

"  If  two  right-angled  triangles  have  the  hypoth- 
ermse  and  a  side  of  the  one  equal  to  the  hypoth- 
enuse  and  a  side  of  the  other,  each  to  each,  the 
triangles   will   be  equal   in   all   their   parts." 

In   the  two   right-angled   triangles  BAC   and  EDF 


GEOMETRY.  121 


(see  next  figure),  let  the  hypothenuse  AC  be  equal 
;o  DF,  the  side  BA  to  the  side  ED :  then  will 
the  side  BC  be  equal  to  EF,  the  angle  A  to  the 
angle  D,  and  the  angle  C  to  the  angle  F.  To  prove 
this  proposition,  we  need  the  following,  which  have 
been  before  proved ;  viz. : 

Prop.  X.  (of  Legendre).  "If  two  triangles  have 
the  three  sides  of  the  one  equal  to  the  three  sides 
of  the  other,  each  to  each,  the  triangles  will  be 
equal  in  all  their  parts." 

Prop.  Y.  "  If  two  triangles  have  two  sides 
and  the  included  angle  of  the  one,  equal  to  two 
sides  and  the  included  angle  of  the  other,  each 
to  each,  the  two  triangles  will  be  equal  in  all  their 
parts." 

Axiom  1.  "  Things  which  are  equal  to  the  same 
thing,  are  equal  to  each  other." 

Axiom  10  (of  Legendre).  u  All  right  angles  are 
equal." 

Prop.  XY.  "  If  from  a  point  without  a  straight 
line,  a  perpendicular  be  let  fall  on  the  line,  and  ob- 
lique lines  be  drawn  to  different  points  of  it, 

1st.  "  The  perpendicular  will  be  shorter  than  any 
oblique  line  ; 

2d.  "  Of  two  oblique  lines,  drawn  at  pleasure,  that 
which  is  farther  from  the  perpendicular  will  be  the 
longer." 

These  are  the  data. 

e 


122  MATHEMATICS. 

Now  the  two  sides  BC  and  EF 
are  either  equal  or  unequal.  If 
they  are  equal,  then  by  Prop.  X. 
the  remaining  parts  of  the  two 
triangles  are  also  equal,  and  the 
triangles  themselves  are  equal  in  all  their  parts.  If 
the  two  sides  are  unequal,  one  of  them  must  be 
greater  than  the  other :  suppose  BC  to  be  the 
greater. 

On  the  greater  side  BO  take  a  part  BG,  equal  to 
EF,  and  draw  AG.  Then,  in  the  two  triangles 
BAG  and  DEF  the  angle  B  is  equal  to  the  angle 
E,  by  axiom  10  (Legendre),  both  being  right  angles. 
The  side  AB  is  equal  to  the  side  DE,  and  by  hypo- 
thesis the  side  BG  is  equal  to  the  side  EF  :  then  it 
follows  from  Prop.  Y.  that  the  side  AG  is  equal  to 
the  side  DF.  But  the  side  DF  is  equal  to  the  side 
AC :  hence,  by  axiom  1,  the  side  AG  is  equal  to 
AC.  But  the  line  AG  cannot  be  equal  to  the  line 
AC,  having  been  shown  to  be  less  than  it  by  Prop. 
XY. ;  hence,  the  conclusion  contradicts  a  known 
truth,  and  is  therefore  false;  consequently,  the  sup- 
position (on  which  the  conclusion  rests),  that  BO 
and  EF  are  unequal,  is  also  false ;  therefore,  as  they 
cannot  be  unequal,  they  are  equal. 

§  174.  It  is  often  supposed,  though  erroneously, 
that  the  Negative  Demonstration,  or  the  demonstra- 


GEOMETRY.  123 


tion  involving  the  "  rednctio  ad  absurdum,"  is  less 
conclusive  and  satisfactory  than  direct  or  positive  de- 
monstration. This  impression  is  simply  the  result  of 
a  want  of  proper  analysis. 

For  example  :  in  the  demonstration  just  given,  it 
was  proved  that  the  two  sides  BC  and  EF  cannot 
be  unequal  /  for,  such  a  supposition,  in  a  logical 
argumentation,  resulted  in  a  conclusion  directly  op- 
posed to  a  known  truth;  and  as  equality  and  in- 
equality are  the  only  general  conditions  of  relation 
between  two  quantities,  it  follows  that  if  they  do 
not  fulfil  the  one,  they  must  the  other.  In  both 
kinds  of  demonstration,  the  premises  and  conclusion 
agree ;  that  is,  they  are  both  true,  or  both  false ; 
and  the  reasoning  or  argument,  in  both,  is  supposed 
to  be  strictly  logical. 

In  the  direct  demonstration,  the  premises  are 
known,  being  antecedent  truths  ;  and  hence,  the  con- 
clusion is  true.  In  the  negative  demonstration,  one 
element  is  assumed,  and  the  conclusion  is  then  com- 
pared with  truths  previously  established.  If  the  con- 
clusion is  found  to  agree  with  any  one  of  these,  we 
infer  that  the  hypothesis  or  assumed  element  is  true; 
if  it  contradicts  any  one  of  these  truths,  we  infer 
that  the  assumed  element  is  false,  and  hence  that  its 
opposite  is  true. 

§  175.  Having  explained  the  meaning  of  the  term 


124  MATHEMATICS 


Measured,  as  applied  to  a  geometrical  magnitude, 
viz.  that  it  implies  the  comparison  of  a  magnitude 
with  its  unit  of  measure ;  and  having  also  explained 
the  signification  of  the  word  Property,  and  the  pro- 
cesses of  reasoning  by  which,  in  all  figures,  proper- 
ties not  before  noticed  are  inferred  from  those  that 
are  known;  we  shall  now  add  a  few  remarks  on  the 
relations  of  the  geometrical  figures,  and  the  methods 
of  comparing  them  with  each  other. 

PROPORTION    OF    FIGURES. 

§  176.  Proportion  is  the  relation  which  one  geomet- 
rical magnitude  bears  to  another  of  the  sa'me  kind, 
with  respect  to  its  being  greater  or  less.  The  two 
magnitudes  so  compared  are  called  terms,  and  the 
measure  of  the  proportion  is  the  quotient  which 
arises  from  dividing  the  second  term  by  the  first, 
and  is  called  their  Ratio.  Only  quantities  of  the 
same  kind  can  be  compared  together,  and  it  follows 
from  the  nature  of  the  relation  that  the  quotient,  or 
ratio,  of  any  two  terms  will  be  an  abstract  num- 
ber, whether  the  terms  themselves  be  abstract  or 
concrete. 

§  177.  The  term  Proportion  is  defined  by  most 
authors,  "An  equality  of  ratios  between  four  num- 
bers or  quantities,  compared  together  two  and  two." 


GEOMETRY.  125 


A  proportion    certainly  arises   from   such   a  compari- 
son :  thus,  if 

A  =  0;then' 
A  :  B  :  :  C  :  D 

is  a  proportion. 

But  if  we  have  two  quantities  A  and  B,  which 
may  change  their  values,  and  are,  at  the  same  time, 
so  connected  together  that  one  of  them  shall  in- 
crease or  decrease  just  as  many  times  as  the  other, 
their  ratio  will  not  be  altered  by  such  changes ; 
and  the  two  quantities  are  then  said  to  be  propor- 
tional. 

Thus,  if  A  be  increased  three  times  and  B  three 
times,  then, 

3  B^A. 
3  A~~B' 

that  is,  3  A  and  3  B  bear  to  each  other  the  same 
proportion,  as  A  and  B.  Science  needs  a  general 
term  to  express  this  relation  between  two  quantities 
which  change  their  values,  without  altering  their 
quotient,  and  the  term  "proportional,"  or  "in  pro- 
portion," is  employed  for  that  purpose. 

There  is  a  symbol  or  character  to  express  the  re- 
lation between  two  quantities,  when  they  undergo 
changes  of  value,  without  altering  their  ratio.     That 


126  MATHEMATICS. 

character  is  a,  and  is  read  "proportional  to."  Thus, 
if  we  have  two  quantities  denoted  by  A  and  B, 
written, 

A  oc  B, 

the  expression  is  read,  u  A  proportional  to  B." 


§  178.  There  is  yet  another  kind  of  relation  which 
may  exist  between  two  quantities  A  and  B,  which  it 
is  very  important  to  consider  and  understand.  Sup- 
pose the  quantities  to  be  so  connected  with  each 
other,  that  when  the  first  is  increased  according  to 
any  law  of  change,  the  second  shall  decrease  accord- 
ing to  the  same  law ;    and  the  reverse. 

For  example  :  the  area  of  a  rectangle 
is  equal  to  the  product  of  its  base  and 
altitude.  Then,  in  the  rectangle  ABCD, 
we  have 

Area=AB  x  BO. 


Take   a  second   rectangle  EFGIT,  having  a  longer 
base  EF,   and   a  less    altitude  FG-, 
but  such  that  it  shall  have  an  equal  H 
area  with  the  first :    then  we  shall 
have 

Area  =  EF  x  FG. 


E 


!N"ow  since  the  areas  are  equal,  we  shall  have 
AB  x  BC  =  EF  x  FG ; 


GEOMETEY.  127 


and  by  resolving  the  terms  of  this  equation  into  a 
proportion,  we  shall  have 

AB  :  EF  :  :  FG  :  BC. 

It  is  plain  that  the  sides  of  the  rectangle  ABCD 
may  be  so  changed  in  value  as  to  become  the  sides 
of  the  rectangle  EFGH,  and  that  while  they  are 
undergoing  this  change,  AB  will  increase  and  BO 
diminish.  The  change  in  the  values  of  these  quan- 
tities will  therefore  take  place  according  to  a  h'xed 
law :  that  is,  one  will  be  diminished  as  many  times 
as  the  other  is  increased,  since  their  product  is  con- 
stantly equal  to  the  area  of  the  rectangle  EFGH. 

Denote  the  side  AB  by  x  and  BC  by  y,  and  the 
area  ofKthe  rectangle  EFGH,  which  is  known,  by  a; 

then 

xy  =  a; 

and  when  the  product  of  two  varying  quantities  is 
constantly  equal  to  a  known  quantity,  the  two  quan- 
tities are  said  to  be  Becvprocally  or  Inversely  pro- 
portional ;  thus  x  and  y  are  said  to  be  inversely  pro- 
portional to  each  other.  If  we  divide  1  bv  each 
member  of  the  above  equation,  we  shall  have 

JL-I  . 

xy  ~  a  ' 

and  by  multiplying  both  members  by  x,  we  shall 
have  1  _  x  * 


128  MATHEMATICS. 

and  then  by  dividing  both  numbers  by  x,  we  have 

•i   i 

x 

that  is,  the  ratio  of  x  to  -  is  constantly  equal  to  - ; 
that  is,  equal  to  the  same  quantity,  however  x  or  y 
may  vary;  for,  a  and  consequently  -  does  not  change. 

Cb 

Hence, 

Two  quantities,  which  may  change  their  values, 
are  reciprocally  or  inversely  proportional,  w/ten  one 
is  proportional  to  unity  divided  dy  the  other,  and 
then  their  product  remains  constant. 

We    express    this    reciprocal    or    inverse    relation 

thus : 

A      1 
Aar 

A  is  said  to  be  inversely  proportional  to  B :  the  sym- 
bols also  express  that  A  is  directly  proportional  to 

j^.     If  we  have 
±> 

we  say,  that  A  is  directly  proportional  to  B,  and  in- 
versely proportional  to  C. 
The  terms  Direct,  Inverse  or  Keciprocal,  apply  to 


GEOMETRY.  129 


the  nature  of  the  proportion,  and  not  to  the  Eatio, 
which  is  always  a  mere  quotient  and  the  measure  of 
proportion.  The  term  Direct  applies  to  all  propor- 
tions in  which  the  terms  increase  or  decrease  to- 
gether ;  and  the  term  Inverse  or  Reciprocal  to  those 
in  which  one  term  increases  as  the  other  decreases. 
They  cannot,  therefore,  properly  be  applied  to  ratio 
without  changing  entirely  its  signification  and  de- 
finition. 

COMPARISON    OF    FIGURES. 

§  179.  In  comparing  geometrical  magnitudes,  by 
means  of  their  quotient,  it  is  not  the  quotient  alone 
which  we  consider.  The  comparison  implies  a  gen- 
eral relation  of  the  magnitudes,  which  is  measured 
by  the  Eatio.  For  example :  we  say  that  "  Similar 
triangles  are  to  each  other  as  the  squares  of  their 
homologous  sides.''  What  do  we  mean  by  that? 
Just  this  : 

That  the   area  of  a  triangle 

Is  to  the  area  of  a  similar  triangle, 

As  the  area  of  a  square  described  on  a  side  of  the 
first 

To  the  area  of  a  square  described  on  a  homolo- 
gous side  of  the  second. 

Thus,  we  see  that  every  term  of  such  a  proportion  is 
in  fact  a  surface,  and  that  the  area  of  a  triangle  in- 
creases or  decreases  much  faster  than  its  sides ;  that  is, 

6* 


» 
130  MATHEMATICS. 

if  we  double  each  side  of  a  triangle,  the  area  will  be 
four  times  as  great :  if  we  multiply  each  side  by  three, 
the  area  will  be  nine  times  as  great ;  or  if  we  divide 
each  side  by  two,  we  diminish  the  area  four  times,  and 
so  on.     Again  : 

The  area  of  one  circle 

Is  to  the  area  of  another  circle, 

As  a  square  described  on  the  diameter  of  the  first 

To  a  square  described  on  the  diameter  of  the 
second. 

Hence,  if"\ve  double  the  diameter  of  a  circle,  the  area 
of  the  circle  whose  diameter  is  so  increased  will  be  in- 
creased four  times;  if  we  multiply  the  diameter  by 
three,  the  area  will  be  increased  nine  times;  and  simi- 
larly, if  we  divide  the  diameter. 

§  180.  In  comparing  volumes  together,  the  same  gen- 
eral principles  obtain.  Similar  volumes  are  to  each 
other  as  the  cubes  described  on  their  homologous  or 
corresponding  sides.     That  is, 

A  prism 

Is  to  a  similar  prism, 

As  a  cube  described  on  a  side  of  the  first 

Is  to  a  cube  described  on  a  homologous  side  of  the 
second. 

Hence,  if  the  sides  of  a  prism  be  doubled,  the  vol- 
umes, or  contents,  will  be  increased  eight-fold. 
Again : 


GEOMETRY.  131 


A  sphere 

Is  to  a  sphere, 

As  a  cube  described  on  the  diameter  of  the  first 

Is  to  a  cube  described  on  a  diameter  of  the 
second. 

Hence,  if  the  diameter  of  a  sphere  be  doubled,  its 
volume,  or  contents,  will  be  increased  eight-fold  ;  if  the 
diameter  be  multiplied  by  three,  the  volume,  or  con- 
tents, will  be  increased  twenty-seven  fold  ;  if  the  diame- 
ter be  multiplied  by  four,  the  volume,  or  contents,  will 
be  increased  sixty -four  fold  ;  the  volumes  increasing  as 
the  cubes  of  the  numbers  1,  2,  3,  4,  &c. 

§  181.  The  relation  or  ratio  of  two  magnitudes  to 
each  other,  may  be,  and  indeed  is,  expressed  by  an  ab- 
stract number.  This  number  has  a  fixed  value  so  long 
as  we  do  not  introduce  a  change  in  the  volumes ;  but, 
if  we  wish  to  express  their  ratio  under  the  supposition 
that  their  volumes  may  change  according  to  fixed  laws 
(that  is,  so  that  the  figures  shall  continue  similar),  we 
then  compare  them  with  similar  figures-  described  on 
their  homologous  or  corresponding  sides ;  or,  what  is 
the  same  thing,  take  into  account  the  corresponding 
changes  which  take  place  in  the  abstract  numbers  that 
express  their  volumes. 


132  MATHEMATICS 


EE  CAPITULATION. 

§  182.  We  have  now  completed  a  general  outline  of 
the  science  of  Geometry,  and  what  has  been  said  may 
be  recapitulated  under  the  following  heads.  It  has 
been  shown, 

1st.  That  Geometry  is  conversant  about  space,  or 
those  limited  portions  of  space  which  are  called,  Geo- 
metrical Magnitudes. 

2d.  That  the  geometrical  magnitudes  embrace  four 
species  of  figures : 

1st.  Lines — straight  and  curved  ; 
2d.  Surfaces — plane  and  curved  ; 
3d.  Volumes — bounded  either  by  plane  surfaces 
or  curved,  or  both ;  and, 

4th.  Angles,  arising  from  the  positions  of  lines 
and  planes,  and  by  which  they  are  bounded. 
3d.  That  the  science  of  Geometry  is  made  up  of  those 
processes  by  means  of  which  all  the  properties  of  these 
magnitudes  are  examined  and  developed,  and  that 
the  results  arrived  at  constitute  the  truths  of  Ge- 
ometry. 

4th.  That  the  truths  of  Geometry  may  be  divided 
into  three  classes; 

1st.  Those  implied  in  the   definitions,  viz.,  that 
things  exist  corresponding  to  certain  words  defined  ; 
2d.  Intuitive  or  self-evident  truths  embodied  in 
the  axioms; 


GEOMETRY.  133 


3d.  Truths  deduced  (that   is,  inferred)  from  the 
definitions     and     axioms,     called     Demonstrative 
Truths. 
5th.  That  the  examination  of  the  properties  of  the 
geometrical  magnitudes  has  reference, 

1st.  To  their  comparison  with  a  standard  or  unit 
of  measure ; 

2d.  To  the  discovery  of  properties  belonging 
to  an  individual  figure,  and  yet  common  to  the 
entire  class  to  which  such  figure  belongs ; 

3d.  To  the  comparison,  with  respect  to  magni- 
tude, of  figures  of  the  same  species  with  each 
other ;  viz.  lines  with  lines,  surfaces  with  surfaces, 
volumes  with  volumes,  and  angles  with  angles. 


SUGGESTIONS   FOR   THOSE   WHO   TEACH    GEOMETRY. 

1.  Be  sure  that  your  pupils  have  a  clear  apprehen- 
sion of  space,  and  of  the  notion  that  Geometry  is  con- 
versant about  space  only. 

2.  Be  sure  that  they  understand  the  signification  of 
the  terms,  lines,  surfaces,  volumes,  and  angles,  and 
that  these  names  indicate  certain  portions  of  space 
corresponding  to  them. 

3.  See  that  they  understand  the  distinction  between 
a  straight  line  and  a  curve ;  between  a  plane  surface 
and  a  curved  surface ;  between  a  volume  bounded  by 


134  MATHEMATICS 


planes  and  a  volume  bounded  by  curved  surfaces ;  and 
also,  between  the  different  kinds  of  angles. 

4.  Be  careful  to  have  them  note  the  characteristics 
of  the  different  species  of  plane  figures,  such  as  tri- 
angles, quadrilaterals,  pentagons,  hexagons,  &c-  ;  and 
then  the  characteristic  of  each  class  or  subspecies,  so 
that  the  name  shall  recall,  at  once,  the  characteristic 
properties  of  each  figure. 

5.  Be  careful,  also,  to  have  them  note  the  character- 
istic differences  of  the  volumes.  Let  them  often  name 
and  distinguish  those  which  are  bounded  by  planes, 
those  bounded  by  plane  and  curved  surfaces,  and 
those  bounded  by  curved  surfaces  only.  Regarding 
Volumes  as  a  genus,  let  them  give  the  species  and 
subspecies  into  which  the  volumes  may  be  divided. 

6.  Having  thus  made  them  familiar  with  the  things 
which  are  the  subjects  of  the  reasoning,  explain  care- 
fully the  nature  of  the  definitions :  then  of  the  axioms, 
the  grounds  of  our  belief  in  them,  and  the  sources 
from  which  those  self-evident  truths  are  inferred. 

7.  Then  explain  to  them,  that  the  definitions  and 
axioms  are  the  basis  of  all  geometrical  reasoning : 
that  every  proposition  must  be  deduced  from  them, 
and  that  they  afford  the  tests  of  all  the  truths  which 
the  reasonings  establish. 

8.  Let  every  figure,  used  in  a  demonstration,  be 
accurately  drawn,  by  the  pupil  himself,  on  a  black- 
board.    This  will  establish  a  connection  between  the 


GEOMETRY.  135 


eye  and  the  hand,  and  give,  at  the  same  time,  a  clear 
perception  of  the  figure  and  a  distinct  apprehension 
of  the  relation  of  its  parts. 

9.  Let  the  pupil,  in  every  demonstration,  first  enun- 
ciate, in  general  terms,  that  is,  without  the  aid  of  a 
diagram,  or  any  reference  to  one,  the  proposition  to 
be  proved  ;  and  then  state  the  principles  previously 
established,  which  are  to  be  employed  in  making  out 
the  proof. 

10.  When,  in  the  course  of  a  demonstration,  any 
truth  is  inferred  from  its  connection  with  one  before 
known,  let  the  truth  so  referred  to  be  fully  and  accu- 
rately stated,  even  though  the  number  of  the  proposi- 
tion in  which  it  is  proved  be  also  required.  This  is 
deemed  important. 

11.  Let  the  pupil  be  made  to  understand  that  a 
demonstration  is  but  a  series  of  logical  arguments 
arising  from  comparison,  and  that  the  result  of  every 
comparison,  in  respect  to  quantity,  contains  the  mark 
either  of  equality  or  inequality. 

12.  Let  the  distinction  between  a  positive  and 
negative  demonstration  be  fully  explained  and  clearly 
apprehended. 

13.  In  the  comparison  of  quantities  with  each  other, 
great  care  should  be  taken  to  impress  the  fact  that 
proportion  exists  only  between  quantities  of  the  same 
kind,  and  that  ratio  is  the  measure  of  proportion. 

14.  Do  not  fail  to  give  much  importance   to  the 


136  MATHEMATICS 


Jcind  of  quantity  under  consideration.  Let  the  ques- 
tion be  often  put,  What  kind  of  quantity  are  you 
considering?  Is  it  a  line,  a  surface,  a  volume,  or  an 
angle  ?  And  what  kind  of  a  line,  surface,  volume  or 
angle  ? 

15.  In  all  cases  of  measurement,  the  unit  of  measure 
should  receive  special  attention.  If  lines  are  measured, 
or  compared  by  means  of  a  common  unit,  see  that 
the  pupil  perceives  that  unit  clearly,  and  apprehends 
distinctly  its  relations  to  the  lines  which  it  measures. 
In  surfaces,  take  much  pains  to  mark  out  on  the  black- 
board the  particular  square  which  forms  the  unit  of 
measure,  and  write  unit,  or  unit  of  measure,  over  it. 
So  in  the  measurement  of  volumes,  let  the  unit  or 
measuring  cube  be  exhibited,  and  the  conception  of 
its  size  clearly  formed  in  the  mind;  and  then  impress 
the  important  fact,  that,  all  measurement  consists  in 
merely  comparing  a  unit  of  measure  with  the  quan- 
tity measured f  and  that  the  number  which  expresses 
the  ratio  is  the  numerical  expression  for  that  measure. 

16.  Be  careful  to  explain  the  difference  of  the  terms 
Equal  and  Equal  in  all  the  parts,  and  never  permit 
the  pupil  to  use  them  as  synonymous.  An  accurate 
use  of  words  leads  to  nice  discriminations  of  thought. 


SECTION  V. 

ANALYSIS. 


§  183.  Analysis  is  a  general  term,  embracing  that 
entire  portion  of  mathematical  science  in  which  the 
quantities  considered  are  represented  by  letters  of 
the  alphabet,  and  the  operations  to  be  performed  on 
them  are  indicated  by  signs. 

§  184.  We  have  seen  that  all  numbers  must  be 
numbers  of  something ;  for,  there  is  no  such  thing 
as  a  number  without  a  base:  that  is,  every  number 
must  be  based  on  the  abstract  unit  one,  or  on  some 
unit  denominated.  But  although  numbers  must  be 
numbers  of  something,  yet  they  may  be  numbers  of 
any  thing,  for  the  unit  may  be  whatever  we  choose 
to  make  it. 

§  185.  All  quantity  consists  of  parts,  which  can  be 
numbered  exactly  or  approximatively,  and,  in  this 
respect,  possesses  all  the  properties  of  number.  Prop- 
ositions, therefore,  concerning   numbers,  have   the   re- 


138  MATHEMATICS 


markable  peculiarity,  that  they  are  propositions  con- 
cerning all  quantities  whatever.  That  half  of  six  is 
three,  is  equally  true,  whatever  the  word  six  may 
represent,  whether  six  abstract  units,  six  men,  or 
six  triangles.  Analysis  extends  the  generalization 
still  further.  A  number  represents,  or  stands  for, 
that  particular  number  of  things  of  the  same  kind, 
without  reference  to  the  nature  of  the  thing;  but  an 
analytical  symbol  does  more,  for  it  may  stand  for 
all  numbers,  or  for  all  quantities  which  numbers  rep- 
resent, or  even  for  quantities  which  cannot  be  ex- 
actly expressed  numerically. 

As  soon  as  we  conceive  of  a  thing  we  may  con- 
ceive it  divided  into  equal  parts,  and  may  represent 
either  or  all  of  those  parts  by  a  or  x,  or  may,  if  we 
please,  denote  the  thing  itself  by  a  or  a?,  without 
any  reference  to  its  being  divided  into  parts. 

§  186.  In  Geometry,  each  geometrical  figure  stands 
for  a  class  :  and  when  we  have  demonstrated  a  prop- 
erty of  a  figure,  that  property  is  considered  as  proved 
for  every  figure  of  the  class.  For  example :  when  we 
prove  that  the  square  described  on  the  hypothenuse 
of  a  right-angled  triangle  is  equal  to  the  sum  of  the 
squares  described  on  the  other  two  sides,  we  demon- 
strate the  fact  for  all  right-angled  triangles.  But  in 
analysis,  all  numbers,  all  lines,  all  surfaces,  all  vol- 
umes,  and   all   angles,  may  be   denoted   by   a   single 


ANALYSIS.  139 


symbol,  a  or  x.  Hence,  all  truths  inferred  by  means 
of  these  symbols  are  true  of  all  things  whatever, 
and  not,  like  those  of  number  and  geometry,  true 
only  of  particular  classes  of  things.  It  is,  therefore, 
not  surprising,  that  the  symbols  of  analysis  do  not 
excite  in  our  minds  the  ideas  of  particular  things. 
The  mere  written  characters,  a,  &,  c,  d,  x,  y,  2,  serve 
as  the  representatives  of  things  in  general,  whether 
abstract  or  concrete,  whether  known  or  unknown, 
whether  finite  or  infinite. 

§  1ST.  In  the  nses  which  we  make  of  these  symbols, 
and  the  processes  of  reasoning  carried  on  by  means 
of  them,  the  mind  insensibly  comes  to  regard  them 
as  things,  and  not  as  mere  signs ;  and  we  constantly 
predicate  of  them  the  properties  of  things  in  general, 
without  pausing  to  inquire  what  kind  of  thing  is  im- 
plied. Thus,  we  define  an  equation  to  be  a  proposi- 
tion in  which  equality  is  predicated  of  one  thing  as 
compared  with  another.     For  example : 

a  +  c  =  x, 

is  an  equation,  because  x  is  declared  to  be  equal  to 
the  sum  of  a  and  c.  In  the  solution  of  equations,  we 
employ  the  axioms,  "  If  equals  be  added  to  equals, 
the  sums  will  be  equal ;"  and,  "  If  equals  be  taken 
from   equals,    the   remainders  will  be  equal."      Now, 


140  MATHEMATICS. 

these  axioms  do  not  express  qualities  of  language, 
but  properties  of  quantity.  Hence,  all  inferences  in 
mathematical  science,  deduced  through  the  instru- 
mentality of  symbols,  whether  Arithmetical,  Geo- 
metrical, or  Analytical,  must  be  regarded  as  con- 
cerning quantity,  and  not  symbols. 

Since  analytical  symbols  are  the  representatives  of 
quantity  in  general,  there  is  no  necessity  of  keeping 
the  idea  of  quantity  continually  alive  in  the  mind ; 
and  the  processes  of  thought  may,  without  danger, 
be  allowed  to  rest  on  the  symbols  themselves,  and 
therefore  become,  to  that  extent,  merely  mechanical. 
But,  when  we  look  back  and  see  on  what  the  rea- 
soning is  based,  and  how  the  processes  have  been 
conducted,  we  shall  find  that  every  step  was  taken 
on  the  supposition  that  we  were  actually  dealing 
with  things,  and  not  symbols ;  and  that,  without 
this  understanding  of  the  language,  the  whole  system 
is  without  signification,  and  fails. 

§  188.  There  are  four  principal  branches  of  Analysis  : 
1st.  Algebra. 

2d.  Analytical  Geometry. 
3d.  Analytical  Trigonometry. 
4th.  Differential  and  Integral  Calculus. 


ANALYSIS.  14:1 


ALGEBEA. 

§  189.  Algebra  is,  in  fact,  a  species  of  universal 
Arithmetic,  in  which  letters  and  signs  are  employed 
to  abridge  and  generalize  all  processes  involving 
numbers.  It  is  divided  into  two  parts,  correspond- 
ing to  the  science  and  art  of  Arithmetic  : 

1st.  That  which  has  for  its  object  the  investiga- 
tion of  the  properties  of  numbers,  embracing  all  the 
processes  of  reasoning  by  which  new  properties  are 
inferred  from  known  ones ;  and, 

2d.  The  solution  of  all  problems  or  questions  in- 
volving the  determination  of  certain  numbers  which 
are  unknown,  from  their  connection  with  certain 
others  which  are  known  or  given. 


ANALYTICAL     GEOMETRY. 

§  190.  Analytical  Geometry  examines  the  proper- 
ties, measures,  and  relations  of  the  geometrical  mag- 
nitudes by  means  of  the  analytical  symbols.  This 
branch  of  mathematical  science  originated  with  the 
illustrious  Descartes,  a  celebrated  French  mathema- 
tician of  the  seventeenth  century.  He  observed 
that  the  positions  of  points,  the  direction  of  lines, 
and  the  forms  of  surfaces,  could  be  expressed  by 
means   of    the   algebraic   symbols ;  and   consequently. 


142  MATHEMATICS. 

iii 

that  every  change,  either  in  the  position  or  extent 
of  a  geometrical  magnitude,  produced  a  correspond- 
ing change  in  certain  symbols,  by  which  such  mag- 
nitude could  be  represented.  As  soon  as  it  was 
found  that,  to  every  variety  of  position  in  points, 
direction  in  lines,  or  form  of  curves  or  surfaces, 
there  corresponded  certain  analytical  expressions 
(called  their  equations),  it  followed,  that  if  the  pro- 
cesses were  known  by  which  these  equations  could 
be  examined,  the  relation  of  their  parts  determined, 
and  the  laws  according  to  which  those  parts  vary, 
relative  to  one  another,  ascertained,  then  the  cor- 
responding changes  in  the  geometrical  magnitudes, 
thus  represented,  could  be  immediately  inferred. 

Hence,  it  follows  that  every  geometrical  question 
can  be  solved,  if  we  can  resolve  the  corresponding 
algebraic  equation  ;  and  the  power  over  the  geomet- 
rical magnitudes  was  extended  just  in  proportion  as 
the  properties  of  quantity  were  brought  to  light  by 
means  of  the  Calculus.  The  applications  of  this  Cal- 
culus were  soon  extended  to  the  subjects  of  mechan- 
ics, astronomy,  and  indeed,  in  a  greater  or  less  degree, 
to  all  branches  of  natural  philosophy;  so  that,  at 
the  present  time,  all  the  varieties  of  physical  phe- 
nomena, with  which  the  higher  branches  of  the 
science  are  conversant,  are  found  to  answer  to  vari- 
eties determinable  by  the  algebraic  analysis. 


ANALYSIS.  143 


Two  classes  of  quantities,  and  consequently  two 
sets  of  symbols,  quite  distinct  from  each  other, 
enter  into  this  Calculus  ;  the  one  called  Constants, 
which  preserve  a  tixed  or  given  value  throughout  the 
same  discussion  or  investigation ;  and  the  other  called 
Variables,  which  undergo  certain  changes  of  value, 
the  laws  of  which  are  indicated  by  the  algebraic 
expressions  or  equations,  into  which  they  enter. 
Hence, 

Analytical  Geometry  may  be  defined  as  that  branch 
of  mathematical  science  which  examines,  discusses, 
and  develops  the  properties  of  the  geometrical  mag- 
nitudes by  noting  the  changes  that  take  place  in  the 
algebraic  symbols  which  represent  them,  the  laws  of 
change  being  determined  by  an  algebraic  equation  or 
formula. 

ANALYTICAL    TRIGONOMETRY. 

§  191.  Analytical  Trigonometry  is  that  branch  of 
Mathematics  which  treats  of  the  general  properties 
and  relations  of  the  sides  and  angles  of  Triangles, 
by  means  of  Analysis. 

DIFFERENTIAL    AND    INTEGRAL    CALCULUS. 

§  192.  In  this  branch  of  mathematical  science,  as 
in  Analytical  Geometry,  two  kinds  of  quantity  are 
considered,   viz.    Variables  and   Constants ;    and,   con- 


144  MATHEMATICS. 

sequently,  two  distinct  sets  of  symbols  are  employed. 
The  science  consists  of  a  series  of  processes  which 
note  the  changes  that  take  place  in  the  values  of  the 
Variables.  Those  changes  of  value  take  place  accord- 
ing to  fixed  laws  established  by  algebraic  formulas, 
and  are  indicated  by  certain  marks  drawn  from  the 
variable  symbols,  called  Differential  Coefficients.  By 
these  marks  we  are  enabled  to  trace  out,  with  the 
accuracy  of  exact  science,  the  most  hidden  properties 
of  quantity,  as  well  as  the  most  general  and  minute 
laws  which  regulate  its  changes  of  value. 

§  193.  It  will  be  observed,  that  Analytical  Geom- 
etry and  the  Differential  and  Integral  Calculus  treat 
of  quantity  regarded  under  the  same  general  aspect, 
viz.  as  subject  to  changes  or  variations  in  magnitude, 
according  to  laws  indicated  by  algebraical  formulas; 
and  the  quantities,  whether  variable  or  constant,  are, 
in  both  cases,  represented  by  the  same  algebraic  sym- 
bols, viz.  the  constants  by  the  first,  and  the  varia- 
bles by  the  final  letters  of  the  alphabet.  There  is, 
however,  this  important  difference:  in  Analytical  Ge- 
ometry all  the  results  are  inferred  from  the  relations 
which  exist  between  the  quantities  themselves,  while 
in  the  Differential  and  Integral  Calculus  they  are 
deduced  by  considering  what  may  be  indicated  by 
marks  drawn  from  variable  quantities,  under  certain 
suppositions,  and  by  marks  of  such  marks. 


ANALYSIS.  145 


§  194.  Algebra,  Analytical  Geometry,  the  Differ- 
ential and  Integral  Calculus,  extended  into  the  Theo- 
ry of  Yariations,  make  up  the  subject  of  analytical 
science,  of  which  Algebra  is  the  elementary  branch. 
As  the  limits  of  this  work  do  not  permit  us  to  dis- 
cuss the  subject  in  full,  we  shall  confine  ourselves  to 
Algebra,  pointing  out,  occasionally,  a  few  of  the 
more  obvious  connections  between  it  and  the  two 
other  branches. 


SECTION    VI 

ALGEBRA. 


§  195.  On  an  analysis  of  the  subject  of  Algebra, 
we  think  it  will  appear  that  the  subject  itself  pre- 
sents no  serious  difficulties,  and  that  most  of  the 
embarrassment  which  is  experienced  by  the  pupil  in 
gaining  a  knowledge  of  its  principles,  as  well  as  in 
their  applications,  arises  from  not  attending  suffi- 
ciently to  the  language  or  signs  of  the  thoughts  which 
are  combined  in  the  reasonings.  At  the  hazard, 
therefore,  of  being  a  little  diffuse,  I  shall  begin  with 
the  very  elements  of  the  algebraic  language,  and 
explain,  with  much  minuteness,  the  exact  significa- 
tion of  the  characters  that  stand  for  the  quantities 
which  are  the  subjects  of  the  analysis;  and  also,  of 
those  signs  which  indicate  the  several  operations  to 
be  performed  on  the  quantities. 

§  196.  The  Quantities  which  are  the  subjects  of 
the  algebraic  analysis  may  be  divided  into  two  classes : 
those  which  are  known  or  given,  and  ^those  which  are 


ALGEBRA.  147 


unknown  or  sought.  The  known  are  uniformly  repre- 
sented by  the  first  letters  of  the  alphabet,  a,  b,  c,  d, 
&c;  and  the  unknown,  by  the  final  letters,  a?,  y,  s,  v, 
w,  &c. 

§  197.  Quantity  is  susceptible  of  being  increased 
or  diminished  and  measured ;  and  there  are  six  oper- 
ations which  can  be  performed  upon  a  quantity  that 
will  give  results  differing  from  the  quantity  itself, 
viz.: 

1st.  To  add  it  to  itself  or  to  some  other  quantity; 

2d.  To  subtract  some  other  quantity  from  it; 

3d.  To  multiply  it  by  a  number; 

4th.  To  divide  it; 

5  th.  To  raise  it  to  any  power ; 

6th.  To  extract  a  root  of  it. 

The  cases  in  which  the  multiplier  or  divisor  is  1, 
are  of  course  excepted ;  as  also  the  cases  where  a  root 
is  to  be  extracted  of  1,  or  1  raised  to  any  power. 

§  198.  The  six  signs  which  denote  these  operations 
are  too  well  known  to  be  repeated  here.  These,  with 
the  signs  of  equality  and  inequality,  the  letters  of  the 
alphabet  and  the  figures  which  are  employed,  make 
up  the  elements  of  the  algebraic  language.  The 
words  and  phrases  of  the  algebraic,  like  those  of 
every  other  language,  are  taken  in  connection  with 
each  other,  and  are  not  to  be  interpreted  as  sepa- 
rate and  isolated   symbols. 


148  MATHEMATICS 


§  199.  The  symbols  of  quantity  are  designed  to 
represent  quantity  in  general,  whether  abstract  or 
concrete,  whether  known  or  unknown;  and  the  signs 
which  indicate  the  operations  to  be  performed  on 
the  quantities  are  to  be  interpreted  in  a  sense  equally 
general.  "When  the  sign  plus  is  written,  it  indicates 
that  the  quantity  before  which  it  is  placed  is  to  be 
added  to  some  other  quantity ;  and  the  sign  minus 
implies  the  existence  of  a  minuend,  from  which  the 
subtrahend  is  to  be  taken.  One  thing  should  be 
observed  in  regard  to  the  signs  which  indicate  the 
operations  that  are  to  be  performed  on  quantities, 
viz.  they  do  not  at  all  affect  or  change  the  nature  of 
the  quantity  before  or  after  which  they  are  written, 
but  merely  indicate  what  is  to  be  done  with  the  quan- 
tity. In  Algebra,  for  example,  the  minus  sign 
merely  indicates  that  the  quantity  before  which  it 
is  written  is  to  be  subtracted  from  some  other 
quantity;  and  in  Analytical  Geometry,  that  the  line 
before  which  it  falls  is  estimated  in  a  contrary  direc- 
tion from  that  in  which  it  would  have  been  reck- 
oned, had  it  had  the  sign  plus;  but  in  neither  case 
is  the  nature  of  the  quantity  itself  different  from 
what  it  would  have  been,  had  it  had  the  sign  plus. 

The  interpretation  of  the  language  of  Algebra  is 
the  first  thing  to  which  the  attention  of  a  pupil 
should  be  directed;  and  he  should  be  drilled  on  the 
meaning   and  import  of  the   symbols,  until  their  sig- 


ALGEBRA.  149 


nifications  and  uses  are  as  familiar  as  the  sounds 
and  combinations  of  the  letters  of  the  alphabet. 

§  200.  Beginning  with  the  elements  of  the  lan- 
guage, let  any  number  or  quantity  be  designated  by 
the  letter  a,  and  let  it  be  required  to  add  this  letter 
to  itself,  and  find  the  result,  or  sum.  The  addition 
will  be  expressed  by 

a  +  a  =  the  sum. 

But  how  is  the  sum  to  be  expressed?  By  simply 
regarding  a  as  one  #,  or  1  a,  and  then  observing 
that  one  a  and  one    a,  make  two  a's  or  2  a :    hence, 

a-\-  a  =2  a] 

and  thus  we  place  a  figure  before  a  letter  to  indicate 
how  many  times  it  is  taken.  Such  figure  is  called 
a   Coefficient. 

§  201.  The  product  of  several  numbers  is  indi- 
cated by  the  sign  of  multiplication,  or  by  simply 
writing  the  letters  which  represent  the  numbers  by 
the  side  of  each  other.      Thus, 

a  x  ft  x  c  x  d  xf,    or    aocdf, 

indicates  the  continued  product  of  a,  ft,  c,  d,  and  f\ 
and   each    letter    is  called  a  factor   of  the  product: 


150  MATHEMATICS 


hence,  a  factor  of  a  product  is  one  of  the  multipliers 
which  produce  it.  Any  figure,  as  5,  written  before 
a  product,  as 

5  a  he  df, 

is  the  coefficient  of  the  product,  and  shows  that  the 
product  is  taken  5  times. 

§  202.  If  the  numbers  represented  by  a,  5,  c,  d,  and 
f  were  equal  to  each  other,  they  would  each  be  rep- 
resented by  a  single  letter  «,  and  the  product  would 
then  become 

axaxaxaxa  =  a5; 

that  is,  we  indicate  the  product  of  several  equal  fac- 
tors by  simply  writing  the  letter  once  and  placing  a 
figure  above  and  a  little  at  the  right  of  it,  to  indicate 
how  many  times  it  is  taken  as  a  factor.  The  figure 
so  written  is  called  an  exponent.  Hence,  an  exponent 
is  a  simple  form  of  expression,  to  point  out  how  many 
equal  factors  are  employed. 

§  203.  To  denote  that  any  quantity  denoted  by  a, 
is  to  be  raised  to  any  power,  as  the  fifth,  for  example, 
we  merely  write  it  with  an  exponent  5 ;  thus, 


which  is  read,  a  to  the  fifth  power. 


ALGEBRA.  151 


§  204.  The  division  of  one  quantity  by  another  is 
indicated  by  simply  writing  the  divisor  below  the  divi- 
dend, after  the  manner  of  a  fraction  ;  by  placing  it  on 
the  right  of  the  dividend  with  a  horizontal  line  and 
two  dots  between  them;  or  by  placing  it  on  the  right 
with  a  vertical  line  between  them  :  thus  either  form 
of  expression, 

— ,  h  -r  tf,   or,   b  |  #, 

Cb 

indicates  the  division  of  h  by  a. 


§  205.  The  extraction  of  a  root  is  indicated  by  the 
sign  sf.  This  sign,  when  used  by  itself,  indicates  the 
lowest  root,  viz.  the  square  root.  If  any  other  root  is 
to  be  extracted,  as  the  third,  fourth,  fifth,  &c,  the 
figure  marking  the  degree  of  the  root  is  written  above 
and  at  the  left  of  the  sign ;  as, 

v   denotes  the  cube  root;   V7~fourth  root,  &c. 

The  figure  so  written,  is  called  the  Index  of  the  root. 
We  have  thus  given  the  very  simple  and  general 
language  by  which  we  indicate  each  of  the  six  opera- 
tions that  may  be  performed  on  an  algebraic  quantity, 
and  every  process  in  Algebra  involves  one  or  other  of 
these  operations. 


152  MATHEMATICS. 


MINUS     SIGN. 

§  206.  The  algebraic  symbols  are  divided  into  two 
classes  entirely  distinct  from  each  other,  viz.  the  sym- 
bols which  represent  quantities,  and  the  signs  which 
denote  operations.  "We  have  seen  that  all  the  algebraic 
processes  are  comprised  under  addition,  subtraction, 
multiplication,  division,  raising  of  powers,  and  the 
extraction  of  roots ;  and  it  is  plain,  that  the  nature 
of  a  quantity  is  not  at  all  changed  by  prefixing  to  it 
the  sign  which  indicates  either  of  these  operations. 
The  quantity  denoted  by  the  letter  a,  for  example,  is 
the  same,  in  every  respect,  whatever  sign  may  be  pre- 
fixed to  it ;  that  is,  whether  it  be  added  to  another 
quantity,  subtracted  from  it,  whether  multiplied  or 
divided  by  any  number,  or  whether  we  extract  the 
square  or  cube  or  any  other  root  of  it.  The  algebraic 
signs,  therefore,  must  be  regarded  merely  as  indicating 
operations  to  be  performed  on  quantity,  and  not  as 
affecting  the  nature  of  the  quantities  to  which  they 
may  be  prefixed.  "We  say,  indeed,  that  quantities  are 
plus  and  minus,  but  this  is  an  abbreviated  language 
to  express  that  they  are  to  be  added  or  subtracted. 

§  207.  In  Algebra,  as  in  Arithmetic  and  Geometry, 
all  the  principles  of  the  science  are  deduced  from  the 
definitions  and  axioms;  and  the  rules  for  performing 


ALGEBRA.  153 


the  operations  are  but  directions  framed  in  conformity 
to  sncli  principles.  Having,  for  example,  fixed  by 
definition  the  power  of  the  minus  sign,  viz.  that  any 
quantity  before  which  it  is  written,  shall  be  regarded 
as  to  be  subtracted  from  another  quantity,  we  wish  to 
discover  the  process  of  performing  that  subtraction,  so 
as  to  deduce  therefrom  a  general  principle,  from  which 
we  can  frame  a  rule  applicable  to  all  similar  cases. 

SUBTRACTION. 

§  208.  Let  it  be  required,  for  example,  to  subtract 
from  b  the  difference  between  a  and  c. 
Now,  having  written  the  letters,  with  their 
proper  signs.,  the  language  of  Algebra  ex- 
presses that  it  is  the  difference  only  between  a  and  c, 
which  is  to  be  taken  from  b;  and  if  this  difference 
were  known,  we  could  make  the  subtraction  at  once. 
But  the  nature  and  generality  of  the  algebraic  sym- 
bols, enable  us  to  indicate  operations,  merely,  and  we 
cannot  in  general  make  reductions  until  we  come  to 
the  final  result.  In  what  general  way,  therefore,  can 
we  indicate  the  true  difference  ? 

If  we  indicate  the  subtraction  of  a  from 
b,  we  have  b  —  a  ;  but  then,  we  have  taken 
away  too'  much  from  b  by  the  number  of 


b—a 

b—  a  +  o 

units  in  c,  for  it  was  not  a,  but  the  difference  between 
a  and  o  that  was  to  be  subtracted  from  5.     Having 

7* 


154  MATHEMATICS, 


taken  away  too  much,  the  remainder  is  too  small  by  c: 
hence,  if  o  be  added,  the  true  remainder  will  be  ex- 
pressed by  b  —  a  +  o. 

Now,  by  analyzing  this  result,  we  see  that  the  sign 
of  every  term  of  the  subtrahend  has  been  changed; 
and  what  has  been  shown  with,  respect  to  these 
quantities  is  equally  true  of  all  others  standing 
in  the  same  relation :  hence,  we  have  the  follow- 
ing general  rule  for  the  subtraction  of  algebraic 
quantities  : 

Change  the  sign  of  every  term  of  the  subtrahend,  or 
conceive  it  to  he  changed,  and  then  unite  the  quan- 
tities as  in  addition. 


MULTIPLICATION. 

§  209.  Let  us  now  consider  the  case  of  multiplica- 
tion, and  let  it  be  required  to  multiply  a  —  b  by  o. 
The  algebraic  language  expresses  that  the  difference 
between  a  and  b  is  to  be  taken  as  many 
times   as    there    are   units   in    c.      If    we  a—b 

knew   this   difference,    we    could    at   once  c 

perform  the  multiplication.  But  by  what 
general  process  is  it  to  be  performed  with- 
out finding  that  difference?  If  we  take  a,  c  times, 
the  product  will  be  ac :  but  as  it  was  only 'the  dif- 
ference between  a  and  b,  that  was  to  be  multiplied 
by  c\  therefore,  this  product  ac  will  be  too  great  by 


ac—bc 


ALGEBRA.  155 


b  taken  c  times ;  that  is,  the  true  product  will  be  ex- 
pressed by  ac—bc.    Hence,  we  see,  that, 

If  a  quantity  having  a  plus  sign  be  multiplied  by 
another  quantity  having  also  a  plus  sign,  the  sign  of 
the  product  will  be  plus ;  and  if  a  quantity  hav- 
ing a  minus  sign  be  midtiplied  by  a  quantity  hav- 
ing a  plus  sign,  the  sign  of  the  product  will  be 
minus.    • 

§  iJlO.  Let  us  now  take  the  most  general  case, 
viz.  that  in  which  it  is  required  to  multiply  a— b  by 
c—d. 

Let  us  again  observe  that  the  algebraic  language 
denotes  that  a  —  b  is  to  be  taken 
as  many  times  as  there  are  units 
in  o  —  d;  and  if  these  two  dif- 
ferences were  known,  their  prod- 
uct would  at  once  form  the  prod- 
uct required. 

First:  let  us  take  a— b  as  many  times  as  there  are 
units  in  o ;  this  product,  from  what  has  already  been 
shown,  is  equal  to  ac—bc.  But  since  the  multiplier 
is  not  c,  but  c—d,  it  follows  that  this  product  is  too 
large  by  a— b  taken  d  times;  that  is,  by  ad—bd: 
hence,  the  first  product  diminished  by  this  last,  will 
give  the  true  product.  But,  by  the  rule  for  subtrac- 
tion, this  difference  is  found   by  changing   the  signs 


a—b 
c—d 


ac—bc 

—  ad  +  bd 


ac—bc— ad  -\-bd 


of  the  subtrahend,  and  then  uniting  all  the  terms 


156  MATHEMATICS 


in  addition  :  hence,  the  true  product  is  expressed  by 
ac—bc—ad  +  bd. 

By  analyzing  this  result,  and  employing  an  ab- 
breviated language,  we  have  the  following  general 
principle  to  which  the  signs  conform  in  multiplica- 
tion, viz. : 

Plus  multiplied  by  plus  gives  plus  in  the  product ; 
plus  multiplied  by  minus  gives  minus  /  minus  multi- 
plied by  plus  gives  minus  f  and  minus  multiplied  by 
minus  gives  plus  in  the  product. 

§  211.  The  remark  is  often  made  by  pupils  that 
the  above  reasoning  appears  very  satisfactory  so  long 
as  the  quantities  are  presented  under  the  above  form; 
but  why  wTill  —  &  multiplied  by  —d  give  plus  bd  ? 
How  can  the  product  of  two  negative  quantities 
standing  alone  be  plus  ? 

In  the  first  place,  the  minus  sign  being  prefixed 
to  b  and  d,  shows  that  in  an  algebraic  sense  they  do 
not  stand  by  themselves,  but  are  connected  with 
other  quantities  :  and  if  they  are  not  so  connected, 
the  minus  sign  makes  no  difference;  for,  it  in  no 
case  affects  the  quantity,  but  merely  points  out  a 
connection  with  other  quantities.  Besides,  the  prod- 
uct determined  above,  being  independent  of  any  par- 
ticular value  attributed  to  the  letters  a,  b,  c,  and  d, 
must  be  of  such  a  form  as  to  be  true  for  all  values; 
and  hence  for   the  case  in  which  a  and  c  are  both 


ALGEBRA.  157 


equal   to   zero.      Making   this   supposition,    the  prod- 
uct reduces  to  the  form  of  +  bd. 

The  rules  for  the  signs  in  division  are  readily  de- 
duced from  the  definition  of  division,  and  the  prin- 
ciples already  laid  down. 


ZERO    AND    INFINITY. 

§  212.  The  terms  zero  and  infinity  have  given  rise 
to  much  discussion,  and  been  regarded  as  presenting 
difficulties  not  easily  removed.  It  may  not  be  easy 
to  frame  a  form  of  language  that  shall  convey  to  a 
mind,  but  little  versed  in  mathematical  science,  the 
precise  ideas  which  these  terms  are  designed  to  ex- 
press ;  but  we  are  unwilling  to  suppose  that  the 
ideas  themselves  are  beyond  the  grasp  of  an  ordinary 
intellect.  The  terms  are  used  to  designate  the  two 
limits  of  each  of  the  quantities,  Space,  Number,  and 
Time. 

§  213.  Assuming  any  two  points  in  space,  and  join- 
ing them  by  a  straight  line,  the  distance  between  the 
points  will  be  truly  indicated  by  the  length  of  this 
line,  and  this  length  may  be  expressed  numerically 
by  the  number  of  times  which  the  line  contains  a 
known  unit. 

If,  now,  the  points  are  made  to  approach  each 
other,   the  length    of  the   line   will   diminish   as  the 


158  MATHEMATICS. 

points  come  nearer  and  nearer  together,  until  at 
length,  when  the  two  points  become  one,  the  length 
of  the  line  will  disappear,  having  attained  its  limits 
which  is  called  zero. 

If,  on  the  contrary,  the  points  recede  from  each 
otner,  the  length  of  the  line  joining  them  will  con- 
tinually increase;  but  so  long  as  the  length  of  the 
line  can  be  expressed  in  terms  of  a  known  unit  of 
measure,  it  is  not  infinite.  But,  if  we  suppose  the 
points  removed,  so  that  any  known  unit  of  measure 
would  occupy  no  appreciable  portion  of  the  line,  then 
the  length  of  the  line  is  said  to  be  Infinite. 

§  214.  Assuming  one  as  the  unit  of  number,  and 
admitting  the  self-evident  truth  that  it  may  be  in- 
creased .or  diminished,  we  shall  have  no  difficulty  in 
understanding  the  import  of  the  terms  zero  and  in- 
finity, as  applied  to  number.  For,  if  we  suppose  the 
unit  one  to  be  continually  diminished,  by  division  or 
otherwise,  the  fractional  units  thus  arising  will  be  less 
and  less;  and  in  proportion  as  we  continue  the  di- 
visions, they  will  continue  to  diminish.  Now,  the 
limit  or  boundary  to  which  these  very  small  fractions 
approach,  is  called  Zero,  or  nothing.  So  long  as  the 
fractional  number  forms  an  appreciable  part  of  one,  it 
is  not  zero,  but  a  finite  fraction ;  and  the  term  zero  is 
only  applicable  to  that  which  forms  no  appreciable 
part  of  the  standard. 


ALGEBRA.  159 


If,  oiTtlie  other  hand,  we  suppose  a  number  to  be 
continually  increased,  the  relation  of  this  number  to 
the  unit  will  be  constantly  changing.  So  long  as  the 
number  can  be  expressed  in  terms  of  the  unit  one,  it  is 
finite,  and  not  infinite;  but  when  the  unit  one  forms 
no  appreciable  part  of  the  number,  the  term  infinite 
is  used  to  express  that  state  of  value,  or,  rather,  that 
limit  of  value. 

§  215.  The  terms  zero  and  infinity  are  there- 
fore employed,  to  designate  the  limits  to  'which  de- 
creasing and  increasing  quantities  may  be  made  to 
approach  nearer  than  any  assignable  quantity;  but 
these  limits  cannot  be  compared,  in  respect  to  magni- 
tude, with  any  known  standard,  so  as  to  give  a  finite 
ratio. 

§  216.  It  may,  perhaps,  appear  somewhat  paradoxi- 
cal, that  zero  and  infinity  should  be  defined  as  "  the 
limits  of  number  and  space"  when  they  are  in  them- 
selves not  measurable.  But  a  limit  is  that  "  which 
sets  bounds  to,  or  circumscribes;"  and  as  all  finite 
space  and  finite  number  (and  such  only  are  implied  by 
the  terms  Space  and  Number),  are  contained  between 
zero  and  infinity,  we  employ  these  terms  to  designate 
the  limits  of  Number  and  Space. 


160  MATHEMATICS. 


OF     THE     EQUATION. 

§  217.  We  have  seen  that  all  deductive  reason  ins: 
involves  certain  processes  of  comparison,  and  that  the 
syllogism  is  the  formula  to  which  those  processes  may 
be  reduced.  It  has  also  been  stated  that  if  two  quan- 
tities be  compared  together,  there  will  necessarily  result 
the  condition  of  equality  or  inequality.  The  equation 
is  an  analytical  formula  for  expressing  equality. 

§  218.  The  subject  of  equations  is  divided  into  two 
parts.  The  first,  consists  in  finding  the  equation;  that 
is,  in  the  process  of  expressing  the  relations  existing  be- 
tween the  quantities  considered,  by  means  of  the  alge- 
braic symbols  and  formula.  This  is  called  the  State- 
ment of  the  proposition.  The  second  is  purely  deduc- 
tive, and  consists,  in  Algebra,  in  what  is  called  the 
solution  of  the  equation,  or  finding  the  value  of  the 
unknown  quantity  :  and  in  the  other  branches  of  analy- 
sis, it  consists  in  the  discussion  of  the  equation  ;  that 
is,  in  the  drawing  out  from  the  equation  every  thing 
which  it  is  capable  of  expressing. 

§  219.  Making  the  statement,  or  finding  the  equa- 
tion, is  merely  analyzing  the  problem,  and  expressing 
its  elements  and  their  relations  in  the  language  of  anal- 
ysis.    It  is,  in  fruth,  collating  the  facts,  noting  their 


ALGEBRA.  161 


bearing  and  connection,  and  inferring  some  general 
law  or  principle  which  leads  to  the  formation  of  an 
equation. 

The  condition  of  equality  between  two  quantities  is 
expressed  by  the  sign  of  equality,  which  is  placed  be- 
tween them.  The  quantity  on  the  left  of  the  sign  of 
equality  is  called  the  first  member,  and  that  on  the 
right,  the  second  member  of  the  equation.  The  first 
member  corresponds  to  the  subject  of  a  proposition ; 
the  sign  of  equality  is  copula  and  part  of  the  predi- 
cate, signifying,  is  equal  to.  Hence,  an  equation  is 
merely  a  proposition  expressed  ^algebraically,  in  which 
equality  is  predicated  of  one  quantity  as  compared 
with  another.     It  is  the  great  formula  of  analysis. 

§  220.  We  have  seen  that  every  quantity  is  either 
abstract  or  denominate ;  hence,  an  equation,  which  is 
a  general  formula  for  expressing  equality,  must  be 
either  abstract  or  denominate. 

An  abstract  equation  expresses  merely  the  relation  of 
equality  between  two  abstract  quantities  :  thus. 

a  +  h  =  x, 

is  an  abstract  equation,  if  no  unit  of  value  be  assigned 
to  either  member;  for,  until  that  be  done  the  abstract 
unit  one  is  understood,  and  the  formula  merely  ex- 


162  MATHEMATICS. 

presses  that  the  sum  of  a  and  h  is  equal  to  a?,  and  is 
true,  equally,  of  all  quantities. 

But  if  we  assign  a  denominate  unit  of  value,  that 
is,  say  that  a  and  b  shall  each  denote  so  many 
pounds  weight,  or  so  many  feet  or  yards  of  length, 
x  will  be  of  the  same  denomination,  and  the  equa- 
tion will  become  denominate. 

§  221.  "We  have  seen  that  there  are  six  operations 
which  may  be  performed  on  an  algebraic  quantity. 
We  assume,  as  an  axiom,  that  if  the  same  operation, 
under  either  of  these  processes,  be  performed  on  both 
members  of  an  equation,  the  equality  of  the  mem- 
bers will  not  be  changed.  Hence,  we  have  the  six 
following 

AXIOMS. 

1.  If  equal  quantities  be  added  to  both  members 
of  an  equation,  the  equality  of  the  members  will  not 
be  destroyed. 

2.  If  equal  quantities  be  substracted  from  both 
members  of  an  equation,  the  equality  will  not  be 
destroyed. 

3.  If  both  members  of  an  equation  be  multiplied 
by  the  same  number,  the  equality  will  not  be  de- 
stroyed. 

4.  If  both  members  of  an  equation  be  divided  by 


ALGEBRA.  163 


the    same     number,    the    equality    will    not    be    de- 
stroyed. 

5.  If  both  members  of  an  equation  be  raised  to 
the  same  power,  the  equality  of  the  members  will 
not  be  destroyed. 

6.  If  the  same  root  of  both  members  of  an  equa- 
tion be  extracted,  the  equality  of  the  members  will 
not  be  destroyed. 

Every  operation  performed  on  an  equation  will  fall 
under  one  or  other  of  these  axioms,  and  they  afford 
the  means  of  solving  all  equations  which  admit  of 
solution. 

§  222.  The  term  Equality,  in  Geometry,  expresses 
that  two  magnitudes  have  equal  measures;  that  is, 
that  they  contain  the  same  unit  an  equal  number  of 
times.     It  has  the  same  signification  in  Algebra. 


G-ENERAL      REMARKS. 

§  223.  We  have  thus  pointed  out  some  of  the 
marked  characteristics  of  analysis.  In  Algebra,  the 
elementary  branch,  the  quantities,  about  which  the 
science  is  conversant,  are  divided,  as  has  been 
already  remarked,  into  known  and  unknown,  and 
the  connections  between  them,  expressed  by  the 
equation,   afford    the  means    of    tracing  out    further 


164  MATHEMATICS. 

relations,  and  of  finding  the  values  of  the  unknown 
quantities,   in  terras  of  the  known. 

In  the  other  branches  of  analysis,  the  quantities 
considered  are  divided  into  two  general  classes,  Con- 
stant and  Variable;  the  former  preserving  fixed  val- 
ues throughout  the  same  process  of  investigation, 
while  the  latter  undergo  changes  of  value  according 
to  fixed  laws;  and  from  such  changes  we  deduce, 
by  means  of  the  equation,  common  principles,  and 
general  properties  applicable  to  all  quantities. 

§  224.  The  correspondence  between  the  processes 
of  reasoning,  as  exhibited  in  the  subject  of  general 
logic,  and  those  which  are  employed  in  mathemat- 
ical science,  is  readily  accounted  for,  when  we  reflect, 
that  the  reasoning  process  is  essentially  the  same  in 
all  cases:  and  that  any  change  in  the  language 
employed,  or  in  the  subject  to  which  the  reasoning 
is  applied,  does  not  at  all  change  the  nature  of  the 
process,  or  materially  vary  its  form. 

§  225.  We  shall  not  pursue  the  subject  of  analysis 
any  further;  for,  it  would  be  foreign  to  the  pur- 
poses of  the  present  work  to  attempt  more  than  to 
point  out  the  general  features  and  characteristics  of 
the  different  branches  of  mathematical  science.  We 
have  aimed  only  to  present  the  subjects  about  which 


ALGEBRA.  165 


the  science  is  conversant,  to  explain  the  peculiari- 
ties of  the  language,  the  nature  of  the  reasoning  pro- 
cesses employed,  and  of  the  connecting  links  of  that 
golden  chain  which  binds  together  all  the  parts, 
forming  an  harmonious  whole. 


SUGGESTIONS   FOR   THOSE   WHO   TEACH   ALGEBRA. 

1.  Be  careful  to  explain  that  the  letters  employed, 
are  the  mere  symbols  of  quantity.  That  of,  and  in 
themselves,  they  have  no  meaning  or  signification 
whatever,  but  are  used  merely  as  the  signs  or  rep- 
resentatives of  such  quantities  as  they  may  be  em- 
ployed to  denote. 

2.  Be  careful  to  explain  that  the  signs  which  are 
used  are  employed  merely  for  the  purpose  of  indica- 
ting the  six  operations  which  may  be  performed  on 
quantity  ;  and  that  they  indicate  operations  merely, 
without  at  all  affecting  the  nature  of  the  quantities 
before  which  they  are  placed. 

3.  Explain  that  the  letters  and  signs  are  the  ele- 
ments of  the  algebraic  language,  and  that  the  sym- 
bolical language  itself  arises  from  the  combination 
of  these  elements. 

4.  Explain  that  the  finding  of  an  algebraic 
formula  is  but  the  translation  of  certain  ideas,  first 
expressed  in  our  common  language,  into  the  language 


166  MATHEMATICS. 

of  Algebra ;  and  that  the  interpretation  of  an  alge- 
braic formula  is  merely  translating  its  various  sig- 
nifications into  common  language. 

5.  Let  the  language  of  Algebra  be  carefully 
studied,  so  that  its  construction  and  significations 
may  be  clearly  apprehended. 

6.  Let  the  difference  between  a  coefficient  and  an 
exponent  be  carefully  noted,  and  the  office  of  each 
often  explained ;  and  illustrate  frequently  the  sig- 
nification of  the  language  by  attributing  numerical 
values  to  letters  in  various  algebraic  expressions. 

T.  Point  out  often  the  characteristics  of  similar 
and  dissimilar  quantities,  and  explain  which  may  be 
incorporated,  and  which  cannot. 

8.  Explain  the  power  of  the  minus  sign,  as  shown 
in  the  four  ground-rules ;  but  very  particularly,  as  it 
is  illustrated  in  subtraction  and  multiplication. 

9.  Point  out  and  illustrate  the  correspondence  be- 
tween the  four  ground-rules  of  Arithmetic  and 
Algebra ;  and  impress  the  fact,  that  their  differ- 
ences, wherever  they  appear,  arise  merely  from  dif- 
ferences in  notation  and  language :  the  principles 
which  govern  the  operations  being  the  same  in  both. 

10.  Explain  with  great  minuteness  and  particular- 
ity all  the  characteristic  properties  of  the  equation  : 
the  manner  of  forming  it ;  the  different  kinds  of 
quantity  which  enter  into  its  composition  ;  its  exam- 


ALGEBRA.  167 


ination    or   discussion ;    and  the  different  methods  of 
elimination. 

11.  In  the  equation  of  the  second  degree,  be  care- 
ful to  dwell  on  the  four  forms  which  embrace  all 
the  cases,  and  illustrate  by  many  examples  that 
every  equation  of  the  second  degree  may  be  re- 
duced to  one  or  other  of  them.  Explain  very  par- 
ticularly the  meaning  of  the  term  root;  and  then 
show,  why  every  equation  of  the  first  degree  has 
one,  and  every  equation  of  the  second  degree  two. 
Dwell  on  the  properties  of  these  roots  in  the  equa- 
tion of  the  second  degree.  Show  why  their  sum, 
in  all  the  forms,  is  equal  to  the  coefficient  of  the 
second  term,  taken  with  a  contrary  sign  ;  and  why 
their  product  is  equal  to  the  absolute  term  with  a 
contrary  sign.  Explain  when  and  wThy  the  roots 
are  imaginary. 

12.  In  fine,  remember  that  every  operation  and 
rule  is  based  on  a  principle  of  science,  and  that  an 
intelligible  reason  may  be  given  for  it.  Find  that 
reason,  and  impress  it  on  the  mind  of  your  pupil 
in  plain  and  simple  language,  and  by  familiar  and 
appropriate  illustrations.  You  will  thus  impress 
right  habits  of  investigation  and  study,  and  he  will 
grow  in  knowledge.  The  broad  field  of  analytical 
investigation  will  be  opened  to  his  intellectual  vision, 
and  he  will  have  made  the  first  steps  in  that  sub- 
lime science  which   discovers   the  laws   of  nature  in 


168  MATHEMATICS. 


their  most  secret  hiding-places,  and  follows  them, 
as  they  reach  out,  in  omnipotent  power,  to  control 
the  motions  of  matter  through  the  entire  regions  of 
occupied  space. 


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